Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$$
Select the correct answer.
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

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
$$h ( x ) = \int _ { 1 } ^ { \sqrt { x } } \frac { 5 z ^ { 2 } } { z ^ { 4 } + 1 } d z$$
Select the correct answer.
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

Quizplus · Accepted Answer

By Part 1 of the Fundamental Theorem of Calculus, we have
$$h'(x) = \frac{d}{dx}\left[\int_{1}^{\sqrt{x}}\frac{5z^2}{z^4+1}dz\right] = \frac{5(\sqrt{x})^2}{(\sqrt{x})^4+1} \cdot \frac{1}{2\sqrt{x}} = \frac{5\sqrt{x}}{2(x^2+1)}.$$
Therefore, the correct choice is $\boxed{\textbf{(C)}}.$

Use Part 1 of the Fundamental Theorem of Calculus to Find