Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$$
A) $\frac { 5 \sqrt { x } } { x ^ { 2 } + 1 }$
B) $\frac { \sqrt { x ^ { 2 } + 1 } } { 5 }$
C)
$\frac { 5 \sqrt { x } } { 2 \left( x ^ { 2 } + 1 \right) }$
D) $\frac { \sqrt { x + 5 } } { x ^ { 2 } + 5 }$
E) none of these

Question

Quizplus · Accepted Answer

Using Part 1 of the Fundamental Theorem of Calculus, we can find the derivative of $h(x)$ by evaluating the integrand at the upper limit of integration (which is $\sqrt{x}$) and then multiplying by the derivative of the upper limit. That is,
$$h'(x) = \frac{5(\sqrt{x})^2}{(\sqrt{x})^4+1}\cdot \frac{d}{dx}(\sqrt{x}) = \frac{5\sqrt{x}}{2(x^2+1)}$$
Therefore, the answer is C.

Use Part 1 of the Fundamental Theorem of Calculus to Find