# How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function g(x) = int(1 / (t^3 + 1)) dt from [1,x]?

May 14, 2015

The theorem says:

Given the function:

$y = {\int}_{h \left(x\right)}^{g} \left(x\right) f \left(t\right) \mathrm{dt}$

then:

$y ' = f \left(g \left(x\right)\right) \cdot g ' \left(x\right) - f \left(h \left(x\right)\right) \cdot h ' \left(x\right)$.

So:

$g ' \left(x\right) = \frac{1}{{x}^{3} + 1} \cdot 1 - \frac{1}{{1}^{3} + 1} \cdot 0 = \frac{1}{{x}^{3} + 1}$.