Fundamental theorem of calculus to find the derivative calculator

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Note that not only the domain of integration but also the integrand depend on $x$. Let's (for a moment) write $F(x) = \int_x^0 f(x, t)\,dt = -\int_0^x f(x,t)\,dt$ for your integral. To handle the "double" $x$-dependence, we write $F$ as the composition of $\Delta\colon \mathbb R \to \mathbb R^2$, $x\mapsto (x,x)$ and $\Phi\colon \mathbb R^2 \to \mathbb R$, $(x_1, x_2) \mapsto -\int_0^{x_1} f(x_2, t)\, dt$. We have $F = \Phi \circ \Delta$, hence $$ F'(x) = \nabla \Phi \bigl(\Delta(x)\bigr) \cdot \Delta'(x) $$ by the chain rule. Now $\Delta'(x) = (1,1)$, and \begin{align*} \partial_1\Phi(x_1, x_2) &= -f(x_2, x_1)\\ \partial_2\Phi(x_1, x_2) &= -\int_0^{x_1} \partial_{x_2} f(x_2, t)\, dt \end{align*} (for the first derivative we used the fundamental theorem). Plugin everything together, we obtain $$ F'(x) = -f(x,x) - \int_0^x \partial_x f(x,t)\, dt $$

answered Jul 5, 2013 at 6:34

martinimartini

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Video transcript

- [Instructor] Let's say that we have the function g of x, and it is equal to the definite integral from 19 to x of the cube root of t dt. And what I'm curious about finding or trying to figure out is, what is g prime of 27? What is that equal to? Pause this video and try to think about it, and I'll give you a little bit of a hint. Think about the second fundamental theorem of calculus. All right, now let's work on this together. So we wanna figure out what g prime, we could try to figure out what g prime of x is, and then evaluate that at 27, and the best way that I can think about doing that is by taking the derivative of both sides of this equation. So let's take the derivative of both sides of that equation. So the left-hand side, we'll take the derivative with respect to x of g of x, and the right-hand side, the derivative with respect to x of all of this business. Now, the left-hand side is pretty straight forward. The derivative with respect to x of g of x, that's just going to be g prime of x, but what is the right-hand side going to be equal to? Well, that's where the second fundamental theorem of calculus is useful. I'll write it right over here. Second fundamental, I'll abbreviate a little bit, theorem of calculus. It tells us, let's say we have some function capital F of x, and it's equal to the definite integral from a, sum constant a to x of lowercase f of t dt. The second fundamental theorem of calculus tells us that if our lowercase f, if lowercase f is continuous on the interval from a to x, so I'll write it this way, on the closed interval from a to x, then the derivative of our capital f of x, so capital F prime of x is just going to be equal to our inner function f evaluated at x instead of t is going to become lowercase f of x. Now, I know when you first saw this, you thought that, "Hey, this might be some cryptic thing "that you might not use too often." Well, we're gonna see that it's actually very, very useful and even in the future, and some of you might already know, there's multiple ways to try to think about a definite integral like this, and you'll learn it in the future. But this can be extremely simplifying, especially if you have a hairy definite integral like this, and so this just tells us, hey, look, the derivative with respect to x of all of this business, first we have to check that our inner function, which would be analogous to our lowercase f here, is this continuous on the interval from 19 to x? Well, no matter what x is, this is going to be continuous over that interval, because this is continuous for all x's, and so we meet this first condition or our major condition, and so then we can just say, all right, then the derivative of all of this is just going to be this inner function replacing t with x. So we're going to get the cube root, instead of the cube root of t, you're gonna get the cube root of x. And so we can go back to our original question, what is g prime of 27 going to be equal to? Well, it's going to be equal to the cube root of 27, which is of course equal to three, and we're done.

Above, enter the function to derive. Differentiation variable and more can be changed in "Options". Click "Go!" to start the derivative calculation. The result will be shown further below.

How the Derivative Calculator Works

For those with a technical background, the following section explains how the Derivative Calculator works.

First, a parser analyzes the mathematical function. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). In doing this, the Derivative Calculator has to respect the order of operations. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write "5x" instead of "5*x". The Derivative Calculator has to detect these cases and insert the multiplication sign.

The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code. MathJax takes care of displaying it in the browser.

When the "Go!" button is clicked, the Derivative Calculator sends the mathematical function and the settings (differentiation variable and order) to the server, where it is analyzed again. This time, the function gets transformed into a form that can be understood by the computer algebra system Maxima.

Maxima takes care of actually computing the derivative of the mathematical function. Like any computer algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known differentiation rules. Maxima's output is transformed to LaTeX again and is then presented to the user.

Displaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. Instead, the derivatives have to be calculated manually step by step. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. In each calculation step, one differentiation operation is carried out or rewritten. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). This, and general simplifications, is done by Maxima. For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible.

The "Check answer" feature has to solve the difficult task of determining whether two mathematical expressions are equivalent. Their difference is computed and simplified as far as possible using Maxima. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. If it can be shown that the difference simplifies to zero, the task is solved. Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places.

The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. While graphing, singularities (e. g. poles) are detected and treated specially. The gesture control is implemented using Hammer.js.

If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail.

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