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Explain how to find the derivative of an integral that has an equation as one of its boundaries.

Question:

Explain how to find the derivative of an integral that has an equation as one of its boundaries.

Fundamental Theorem of Calculus:

If we apply the Fundamental Theorem of Calculus we can obtain the derivative of a function that is presented as an integral whose lower limit of integration is a constant and the upper limit has an equation. This theorem is a basic mathematical reference to calculate the derivative of the function under the sign of the integral whose integrand is {eq}f(t) {/eq}, so its derivative is the function {eq}f(x) {/eq} multiplied by the internal derivative of the function that is at the limit of the integral.

Answer and Explanation:

{eq}\eqalign{ & {\text{If the }}\,f{\text{ function is continuous and integrable in an interval }}\left[ {a,b} \right]{\text{, if }}\,y = F\left( x \right)\,{\text{ is defined }} \cr & {\text{by the following integral: }}\,F\left( x \right) = \int\limits_a^{g\left( x \right)} {f\left( t \right)} dt{\text{. Then the derivative of the function }}\,y = F\left( x \right){\text{ }} \cr & {\text{using the Fundamental Theorem of Calculus is calculated as follows:}} \cr & \,\,\,\,\boxed{{F^\prime }\left( x \right) = \frac{d}{{dx}}\left( {\int\limits_a^{g\left( x \right)} {f\left( t \right)} dt} \right) = f\left( {g\left( x \right)} \right) \cdot {g^\prime }\left( x \right)} \cr} {/eq}


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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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