Question

How to find the derivative of an integral with bounds.

Answer 1

To find the derivative of an integral with bounds, use the Fundamental Theorem of Calculus. Find the antiderivative, evaluate it at the bounds, and subtract the results. Then, take the derivative of the resulting expression.

Step-by-step explanation:

To find the derivative of an integral with bounds, you can use the Fundamental Theorem of Calculus.

Here are the steps:

First, find the antiderivative of the integrand function.

Next, evaluate the antiderivative at the upper bound and subtract the result evaluated at the lower bound.

Finally, take the derivative of the resulting expression with respect to the variable of integration.

Let's denote the lower bound as 'a' and the upper bound as 'b', and let the integrand function be denoted by 'f(x)'. Then, the derivative of the integral is given by d/dx ∫(a to b) f(x) dx = f(b) - f(a).

Answer 2

The statement of the fundamental theorem of calculus shows the upper limit of the integral as exactly the variable of differentiation. Using the chain rule in combination with the fundamental theorem of calculus we may find derivatives of integrals for which one or the other limit of integration is a function of the variable of differentiation.

Step-by-step explanation: