- Free partial derivative calculator - partial differentiation solver step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy
- A partial Derivative Calculatoris a tool which provides you the solution of partial derivate equationssolution with so much ease and fun. It is used to take the equations of derivative or two variables and even it intakes multivariable. That might be the reason why people call it multi-derivativeinstead of partial derivative
- Partial Derivative Calculator is a free online tool that displays the partial derivative for the given function. BYJU'S online partial derivative calculator tool makes the calculation faster, and it displays the partial derivative of a given function in a fraction of seconds. How to Use the Partial Derivative Calculator
- Partial Derivative Calculator This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration
- Partial Derivative Calculator computes derivatives of a function with respect to given variable utilizing analytical differentiation and displays a step-by-step solution. It gives chance to draw graphs of the function and its derivatives. Calculator maintenance derivatives up to 10th order, as well as complex functions
- Get the free Partial Derivative Calculator widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha

- Therefore, partial derivatives are calculated using formulas and rules for calculating the derivatives of functions of one variable, while counting the other variable as a constant. This calculator calculates the derivative of a function and then simplifies it
- The partial derivative calculator on this page computes the partial derivative of your inputted function symbolically with a computer algebra system, all behind the scenes. The computer algebra system is very powerful software that can logically digest an equation and apply every existing derivative rule to it in order
- e two variable function z = f (x, y). Partial derivative by variables x and y are denoted as ∂ z ∂ x and ∂ z ∂ y correspondingly
- The Derivative Calculator supports computing first, second, , fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions
- This is a second order partial derivative calculator. A partial derivative is a derivative taken of a function with respect to a specific variable. The function is a multivariate function, which normally contains 2 variables, x and y. However, the function may contain more than 2 variables
- Partial Derivative Calculator A step by step partial derivatives calculator for functions in two variables. You may first want to review the rules of differentiation of functions and the formulas for derivatives

Enter the function you want to find the derivative of in the editor. The Derivative Calculator supports solving first, second...., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. You can also get a better visual and understanding of the function by using our graphing tool The partial derivative of a multi-variable expression with respect to a single variable is computed by differentiating the given function w.r.t. the desired variable whilst treating all other variables as constant, unlike the total differential where all variables can vary Fractional Partial Derivative. Log InorSign Up. This is an example of pushing the limits of the calculator. f(x,y) is any 3-d function. Try changing it! 1. f x, y = x 2 + y 2. 2. i x, y = 1 1 2 ! 2 · d dx ∫ x −. 0 0 0 0 1 0 x − t 2 − 1 2 · d dy ∫ y −. 0 0 0 0 1 0 y −. Calculate Partial derivatives with incredible ease! Use this handy partial derivative calculator with a step-by-step solution and graph

* Partial Derivative Calculator*. Click here for* Partial Derivative Calculator*. This is a partial derivative calculator. A partial derivative is a derivative taken of a function with respect to a specific variable. The function is a multivariate function, which normally contains 2 variables, x and y. However, the function may contain more than 2. Partial Fractions Calculator - find the partial fractions of a fractions step-by-step. This website uses cookies to ensure you get the best experience. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series

- Free partial derivative calculator - partial differentiation solver step-by-ste
- For the partial derivative with respect to h we hold r constant: f' h = π r 2 (1)= π r 2 (π and r 2 are constants, and the derivative of h with respect to h is 1) It says as only the height changes (by the tiniest amount), the volume changes by π r 2 It is like we add the thinnest disk on top with a circle's area of π r 2
- The partial derivative at (0, 0) must be computed using the limit definition because f is defined in a piecewise fashion around the origin: f (x, y) = (x 3 + x 4 − y 3) / (x 2 + y 2) except that f (0, 0) = 0. You can change the point (x, y) at which ∂ f ∂ x (x, y) is evaluated by dragging the blue point
- Abramowitz and Stegun (1972) give finite difference versions for partial derivatives. A differential equation expressing one or more quantities in terms of partial derivatives is called a partial differential equation. Partial differential equations are extremely important in physics and engineering, and are in general difficult to solve
- In addition, remember that anytime we compute a
**partial****derivative**, we hold constant the variable(s) other than the one we are differentiating with respect to. Activity 10.3.2. Find all second order**partial****derivatives**of the following functions. For each**partial****derivative**you calculate, state explicitly which variable is being held constant - The partial derivative calculator on this web page calculates the partial derivative of your inputted function symbolically with a computer system algebra system, all behind the scenes. The computer system algebra system is a compelling software application that can practically digest formula and also use every existing derivative regulation to.
- In this section we will the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice (i.e. without the use of the definition). As you will see if you can do derivatives of functions of one variable you won't have much of an issue with partial derivatives

** Each of these partial derivatives is a function of two variables, so we can calculate partial derivatives of these functions**. Just as with derivatives of single-variable functions, we can call these second-order derivatives, third-order derivatives, and so on. In general, they are referred to as higher-order partial derivatives Derivative Calculator This simple and convenient derivative calculator will help you solve any problem, just enter the value of the function and you will immediately get a solution with a detailed step-by-step description. The easy-to-use calculator interface allows you to quickly calculate any functions. Try this handy derivative calc right now When a function has a multidimensional input, and a multidimensional output, you can take its partial derivative by computing the partial derivative of each component in the output. Created by Grant Sanderson. Google Classroom Facebook Twitte Utilize our Partial Derivative Calculator tool and obtain the result instantly. Provide input function in the input fields and click on the calculate button to get the partial derivative of the given function along with the detailed solution. Partial Derivative Calculator: Are you scared of finding the partial derivatives Partial Derivative Calculator . Online Calculators with Steps (Free) Texas Instruments TI 84 Calculators; TI-84 Online Calculator ; TI-84 App Download; TI-84 Emulator ; Online Virtual TI-84 Graphing Calculator ; TI-84 Manual (User Guide) Graphing Calculator Online TI-84 ; TI-84 Online

- Free mixed partial derivative calculator - mixed partial differentiation solver step-by-ste
- Enter Function: Differentiate with respect to: Enter the Order of the Derivative to Calculate (1, 2, 3, 4, 5)
- Derivative Calculator gives step-by-step help on finding derivatives. This calculator is in beta. We appreciate your feedback to help us improve it. Please use this feedback form to send your feedback. Thanks
- Derivatives of inverse functions calculator is an alternate to those manual calculations as derivative inverse calculator saves your time you spend doing manual calculations. It is used to increase the productivity and efficiency while learning
- This calculator, makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number. Partial Derivative Calculator

Abramowitz and Stegun (1972) give finite difference versions for **partial** **derivatives**. A differential equation expressing one or more quantities in terms of **partial** **derivatives** is called a **partial** differential equation. **Partial** differential equations are extremely important in physics and engineering, and are in general difficult to solve This is a great Calculus app, with it you can calculate any derivative of any function, you can calculate single, double or triple integrals, you can draw slope fields and you can calculate partial fractions Please check it out! calculus_formulas_83p.zip: 1k: 04-02-18: Basic Formulas of Calculus It shows basic formulas for Calculus ** Get more lessons like this at http://www**.MathTutorDVD.comHere you will learn how to calculate partial derivatives in calculus using the TI-89 graphing calcul.. Definition: Partial derivatives are defined as derivatives of multiple variables when they are fixed during the differentiation, except for the variable of interest.. Let f(x, y) be a two-variable function. If we maintain and differentiate f in relation to the variable x (assuming that f is differentiable), using the rules and formulas for differentiation, we get what is called the' f.

Partial Derivatives Calculator This Maplet serves as a calculator for partial derivatives of functions of two variables. This Maplet was used in a Survey of Calculus and Multivariable Calculus courses. Students can use this Maplet to check their homework or any partial work that requires the computation of partial derivatives Second Order Partial Derivatives in Calculus. Examples with detailed solutions on how to calculate second order partial derivatives are presented. Definitions and Notations of Second Order Partial Derivatives For a two variable function f(x , y), we can define 4 second order partial derivatives along with their notations.. The parentheses are in place to indicate how I broke up the variables to take the derivatives. Now let's calculate the last derivative, the partial derivative with respect to y. fzxy =(2z)(e xyz 2)+(2zy)(xz2e 2)+(2xy2z3)(xz2exyz2)+(4xyz3)(exyz2) After we simplify, we get the ﬁnal answer fzxy =2zexyz 2 ⇥ 1+3xyz2 +x 2y z4 ⇤ Example 5.3.0.8 3 Find the partial derivative f_x (1,2) and relate its value to the sketch you just made. Write the trace f (1,y) at the fixed value x=1\text {.} On the right side of Figure 10.2.5, draw the graph of the trace with x=1 indicating the scale and labels on the axes. Also, sketch the tangent line at the point y=2\text {. where - the derivative of the parametric equation y (t) by the parameter t and - the derivative of the parametric equation x (t), by the parameter t. Our online calculator finds the derivative of the parametrically derined function with step by step solution

- Use the limit definition of partial derivatives to calculate for the function Then, find and by setting the other two variables constant and differentiating accordingly. We first calculate using (Figure), then we calculate the other two partial derivatives by holding the remaining variables constant
- The partial derivative calculator is a very small piece of software. We can download the executable file from the website. Once you download the executable file you can double click the program and it starts the installation and setup process. It will ask a few questions which need to be answered properly with care
- In addition, remember that anytime we compute a partial derivative, we hold constant the variable(s) other than the one we are differentiating with respect to. Activity 10.3.2. Find all second order partial derivatives of the following functions. For each partial derivative you calculate, state explicitly which variable is being held constant
- Partial derivation can also be calculated using the partial derivative calculator above. Derivative formula. Below, you will find the basic and advance derivative rules, which will help you understand the whole process of derivation. Sum rule (af + βg)' = af' + βg' Constant rule

Derivatives of Expressions with Several Variables. To differentiate an expression that contains more than one symbolic variable, specify the variable that you want to differentiate with respect to. The diff command then calculates the partial derivative of the expression with respect to that variable. For example, given the symbolic expressio Implicit Differentiation Calculator online with solution and steps. Detailed step by step solutions to your Implicit Differentiation problems online with our math solver and calculator. Solved exercises of Implicit Differentiation ** Laplace's equation (a partial differential equationor PDE) in Cartesian co-ordinates is u xx+ u yy= 0**. (20) We would like to transform to polar co-ordinates. In the handout on the chain rule (side 2) we found that the xand y-derivatives of utransform into polar co-ordinates in the following way: u x= (cosθ)u r− sinθ r u θ u y= (sinθ)u. Partial Derivative Calculator: Are you scared of finding the partial derivatives? To help you in this we have given the free Partial Derivative Calculator that does all your derivative calculations in fraction of seconds. We have included the step by step procedure on how to solve the partial differential equation

- As these examples show, calculating a partial derivatives is usually just like calculating an ordinary derivative of one-variable calculus. You just have to remember with which variable you are taking the derivative. Example 1 Let f (x, y) = y 3 x 2
- partial-derivative-calculator. pt. Related Symbolab blog posts. High School Math Solutions - Derivative Calculator, Products & Quotients . In the previous post we covered the basic derivative rules (click here to see previous post). We are now going..
- A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant. Example: f (x,y) = x4 + x * y4 Let's partially differentiate the above derivatives in Python w.r.t x import sympy as sy
- Since u₂ has two parameters, partial derivatives come into play. To calculate the derivative of this function, we have to calculate partial derivative with respect to x of u₂(x, u₁). Here, a change in x is reflected in u₂ in two ways: as an operand of the addition and as an operand of the square operator

Partial derivatives are computed similarly to the two variable case. For example, @w=@x means diﬁerentiate with respect to x holding both y and z constant and so, for this example, @w=@x = sin(y + 3z). Note that a function of three variables does not have a graph. 0.7 Second order partial derivatives Again, let z = f(x;y) be a function of x. 11 Partial derivatives and multivariable chain rule 11.1 Basic deﬁntions and the Increment Theorem One thing I would like to point out is that you've been taking partial derivatives all your calculus-life. When you compute df /dt for f(t)=Cekt, you get Ckekt because C and k are constants. The notation df /dt tells you that t is the variable What is the partial derivative, how do you compute it, and what does it mean? Google Classroom Facebook Twitter. Email. Partial derivative and gradient (articles) Introduction to partial derivatives. This is the currently selected item. Second partial derivatives. The gradient

partial-derivative-calculator. he. Related Symbolab blog posts. High School Math Solutions - Derivative Calculator, Products & Quotients . In the previous post we covered the basic derivative rules (click here to see previous post). We are now going.. * The difference between a partial and a complete derivative is what the derivative is with respect to*. This is the same as for an integral---i.e. the reverse of a complete derivative would be an integral over all variables, while the reverse of a partial derivative would be an integral over only the one variable in question We connect each letter with a line and each line represents a partial derivative as shown. Note that the letter in the numerator of the partial derivative is the upper node of the tree and the letter in the denominator of the partial derivative is the lower node of the tree

** Section 3: Higher Order Partial Derivatives 9 3**. Higher Order Partial Derivatives Derivatives of order two and higher were introduced in the package on Maxima and Minima. Finding higher order derivatives of functions of more than one variable is similar to ordinary diﬀerentiation. Example 4 Find ∂2z ∂x2 if z = e(x3+y2) Total Derivative. There are at least two meanings of the term total derivative in mathematics. The first is as an alternate term for the convective derivative.. The total derivative is the derivative with respect to of the function that depends on the variable not only directly but also via the intermediate variables .It can be calculated using the formul So the first derivative of f, from R 3 to R is a 3 by 1 matrix or vector- the gradient vector, in fact. And since the first derivative is from R 3 to R 3, the second derivative is a linear transformation from R 3 to R 3 - which, of course, can be represented by a 3 by 3 matrix- the Hessian that Ray Vickerson mentions Microsoft Mathematics is a good software that can be used as a derivative calculator. It features a partial derivative calculator which can be used as a derivative solver and second derivative calculator.If required, you can also choose to calculate higher order derivatives.You can also view the steps being followed to find the derivative of a function In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).Partial derivatives are used in vector calculus and differential geometry.. The partial derivative of a function (

A partial derivative of a multivariable function is the rate of change of a variable while holding the other variables constant. For a function z = f(x,y), we can take the partial derivative with respect to either x or y. Partial.. Partial Derivatives. Note the two formats for writing the derivative: the d and the ∂. When the dependency is one variable, use the d, as with x and y which depend only on u.The ∂ is a partial.

Partial Derivatives Examples And A Quick Review of Implicit Diﬀerentiation Given a multi-variable function, we deﬁned the partial derivative of one variable with respect to another variable in class. All other variables are treated as constants. Here are some basic examples: 1. If z = f(x,y) = x4y3 +8x2y +y4 +5x, then the partial. Free Summation Calculator. The free tool below will allow you to calculate the summation of an expression. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided How to calculate partial derivatives of multivariable functions in terms of three or more variables . Take the course Want to learn more about Calculus 3? I have a step-by-step course for that. :) Learn More . Finding partial derivatives for three- and four-variable functions 4.3.1 Calculate the partial derivatives of a function of two variables. 4.3.2 Calculate the partial derivatives of a function of more than two variables. 4.3.3 Determine the higher-order derivatives of a function of two variables. 4.3.4 Explain the meaning of a partial differential equation and give an example Partial Derivative Practice Questions. 1. The function f(x, y) gives us the profit (in dollars) of a certain commodity as the number of commodities x sold and the number of days y the commodity is.

Title: Partial Derivative Calculator Reviewed by Rating: 5.0 Summary: Description: You can use this calculator to find partial derivatives, mixed partials and evaluate derivatives at a point. It produces output formatted like a textbook Download Partial Derivative Calculator apk 1.0 for Android. Enjoy millions of the latest Android apps, games, music, movies, TV, books, magazines & more. Anytime.

- The derivative calculator gives chance testing the solutions to calculus exercises. It shows the full working process. The Derivative Calculator helps calculating first, second, fifth derivatives as well as differentiating functions with many variables, implicit differentiation and counting roots, and zeros. There is an opportunity to test the answers
- The derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable. derivative_calculator online. Description : Differentiation calculator. The derivative calculator allows to do symbolic differentiation using the derivation property on one hand and the derivatives of the other usual functions
- Learn about derivatives using our free math solver with step-by-step solutions

The calculator displays the derivative at the bottom of the screen. You can get the derivative at other points, if you need to. Press [2nd F4 makes CALC] [6] again, enter the new x value, and press [ENTER]. What's New. 7 Nov 2020: Converted from HTML 4.01 to HTML5, and italicized variable names Thank you sir for your answers. Actually I need the analytical derivative of the function and the value of it at each point in the defined range. i.e. diff (F,X)=4*3^(1/2)*X; is giving me the analytical derivative of the function. After finding this I also need to find its value at each point of X( i.e., for X=(-1:2/511:+1). Similarly the others Browse other questions tagged derivatives partial-derivative or ask your own question. The Overflow Blog Vote for Stack Overflow in this year's Webby Awards

Free Gradient calculator - find the gradient of a function at given points step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy * 3*.2 Higher Order Partial Derivatives If f is a function of several variables, then we can ﬁnd higher order partials in the following manner. Definition. If f(x,y) is a function of two variables, then ∂f ∂x and ∂f ∂y are also functions of two variables and their partials can be taken. Hence we ca Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchang Partial Derivatives Single variable calculus is really just a special case of multivariable calculus. For the function y = f(x), we assumed that y was the endogenous variable, x was the exogenous variable and everything else was a parameter. For example, given the equation

Higher order partial and cross partial derivatives The story becomes more complicated when we take higher order derivatives of multivariate functions. The interpretation of the first derivative remains the same, but there are now two second order derivatives to consider The partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants. Example. The partial derivative of 3x 2 y + 2y 2 with respect to x is 6xy Derivatives >. A partial derivative is a derivative where one or more variables is held constant.. When you have a multivariate function with more than one independent variable, like z = f (x, y), both variables x and y can affect z.The partial derivative holds one variable constant, allowing you to investigate how a small change in the second variable affects the function's output

- The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. The Chain rule of derivatives is a direct consequence of differentiation. Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions
- g the tangent.
- The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus
- The good news is that, even though this looks like four second-order partial derivatives, it's actually only three. That's because the two second-order partial derivatives in the middle of the third row will always come out to be the same
- Partial Derivative Questions and Answers. Get help with your Partial derivative homework. Access the answers to hundreds of Partial derivative questions that are explained in a way that's easy for.

Calculate Derivative Functions in Python. By Suyash pratap Singh. In this tutorial, we will learn about Derivative function, the rate of change of a quantity y with respect to another quantity x is called the derivative or differential coefficient of y with respect to x. Application of partial derivative: Derivatives in chemistry: One use. An alternative notation is to use escpdesc which gives a partial derivative; thus, typing escpdesc ctrl-t followed by f[x,t] will give the derivative of f with respect to its second argument. For instance, this is a valid way to specify a differential equation High School Math Solutions - Derivative Calculator, the Chain Rule In the previous posts we covered the basic derivative rules, trigonometric functions, logarithms and exponents ( click here ). But we are still missing the most important rule dealing with compound functions, the chain rule We can transform each of these partial derivatives, and others derived in later steps, to two other partial derivatives with the same variable held constant and the variable of differentiation changed. The transformation involves multiplying by an appropriate partial derivative of \(T\), \(p\), or \(V\) Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then ∂ ∂x f(x,y) is deﬁned as the derivative of the function g(x) = f(x,y), where y is considered a constant. It is called partial derivative of f with respect to x. The partial derivative with respect to y is deﬁned similarly. We also use the short hand notation.

derivative of a function with respect to that parameter using the chain rule. • The formulas for calculating such derivatives are dz dt = @f @x dx dt + @f @y dy dt and @z @t = @f @x @x @t + @f @y @y @t • To calculate a partial derivative of a variable with respect to another requires im-plicit di↵erentiation @z @x = Fx Fz, @z @y = Fy F where in the computation of the first partial derivative we have used the identity The Chain Rule for Functions of More than Two Variables We may of course extend the chain rule to functions of n variables each of which is a function of m other variables. This is most easily illustrated with an example First-Order Partial Derivatives Higher-Order Partial Derivatives Exercises Second-Order Partial Derivatives Suppose that we have a function f(x;y) (though the number of variables is irrelevant). Its derivative f x(x;y) is also a function of two variables, so we can often take its partial derivatives, too! Here's how we denote f' Lesson 21: Partial Derivatives in Economics 1. Lesson 21 (Sections 15.6-7) Partial Derivatives in Economics Linear Models with Quadratic Objectives Math 20 November 7, 2007 Announcements Problem Set 8 assigned today. Due November 14. No class November 12

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry Suppose the partial derivatives and both exist. Let denote the product of the functions. Then, we have: generic point, named functions : Suppose are both functions of variables . Then, for any fixed in : These hold wherever the right side expressions make sense (see concept of equality conditional to existence of one side) **Partial** **Derivatives** Examples And A Quick Review of Implicit Diﬀerentiation Given a multi-variable function, we deﬁned the **partial** **derivative** of one variable with respect to another variable in class. All other variables are treated as constants. Here are some basic examples: 1. If z = f(x,y) = x4y3 +8x2y +y4 +5x, then the **partial**. The second derivative calculator is a tool which helps to find the derivative of the derivative of function f. This online calculator makes the calculations easy and enjoyable. Also the problems can be solved instantly. It also helps in differentiating functions with many variables or partial derivatives

- Section 6 Use of Partial Derivatives in Economics; Some Examples Marginal functions. Given that the utility function \(u = f(x,y)\) is a differentiable function and a function of two goods, \(x\) and \(y\): Marginal utility of \(x\), \(MU_{x}\), is the first order partial derivative with respect to \(x\) And the marginal utility of \(y\), \(MU_{y}\), is the first order partial derivative with.
- Fortunately, the concept of implicit differentiation for derivatives of single variable functions can be passed down to partial differentiation of functions of several variables. Suppose that we wanted to find $\frac{\partial z}{\partial x}$
- The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k. By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to R p
- The mixed derivative (also called a mixed partial derivative) is a second order derivative of a function of two or more variables. Mixed refers to whether the second derivative itself has two or more variables. For example: f xy and f yx are mixed,; f xx and f yy are not mixed.; Mixed Derivative Example. The function of two variables f(x, y) can be differentiated with respect to x or y.
- ima. As you learn about partial derivatives you should keep the first point, that all derivatives measure rates of change, firmly in

This section provides an overview of Unit 2, Part C: Lagrange Multipliers and Constrained Differentials, and links to separate pages for each session containing lecture notes, videos, and other related materials

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