Find each derivative using Part I of the Fundamental Theorem of Calculus. (d)/(d x)[∫x^5 sin(t^2

step 1 In order to use part one of the fundamental theorem of calculus, we need the upper limit of inte

Find each derivative using Part I of the Fundamental Theorem...

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Key Concepts

Fundamental Theorem of Calculus (Part I)

This theorem states that if a function is continuous on an interval, then the integral of the function with a variable upper limit defines an antiderivative. In other words, if F(x) = ?[a to x] f(t) dt, where f is continuous, then F?(x) = f(x). This connection between differentiation and integration forms the basis for calculating derivatives of such integral functions.

Reversal of Integration Limits

When the variable appears as the lower limit of an integral rather than the upper limit, the integral can be rewritten by reversing the limits, which introduces a negative sign. Specifically, ?[x to b] f(t) dt is equivalent to -?[b to x] f(t) dt. This manipulation allows the direct application of the Fundamental Theorem of Calculus by shifting the variable dependence to the upper limit.

Chain Rule

The chain rule is a method for computing the derivative of a composite function. In contexts where the integration limits are functions of the variable, the chain rule is applied to correctly account for the derivative of these variable limits. In this problem, although the chain rule is not explicitly needed beyond recognizing the sign change from reversing the integration limits, it is a fundamental tool in more complex variations.

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