# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function: g(s) =...

## Question:

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:

{eq}g(s) = \int_{8}^{s} (t - t^5)^6 dt {/eq}

## The Fundamental Theorem of Calculus

The fundamental Theorem of Calculus Part I states that

{eq}\displaystyle \frac{d}{dx}\left(\int_a^x f(t)dt\right) =f(x), \text{ where } a - \text{ is a constant}. {/eq}

To apply FTC Part I, we need to make sure the upper limit of integration is the variable with respect to which the derivative is taken and

the lower limit of integration is a constant.

To find the derivative of {eq}\displaystyle g(s) = \int_{8}^{s} (t - t^5)^6 dt {/eq}, where the lower limit of integration is a constant, 8

and the upper limit is the variable s, with respect to which the derivative is taken,

we apply the fundamental Theorem of Calculus Part I:

{eq}\displaystyle g'(s)=\frac{d}{ds}\left(\int_{8}^{s} (t - t^5)^6 dt \right) = \boxed{(s - s^5)^6 }, {/eq}

which is the integrand function evaluated at s. 