The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

A) $x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
B) $- x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
C) $2 x \left( 5 ^ { x ^ { 2 } } \right)$
D) $- 2 x \left( 5 ^ { x ^ { 2 } } \right)$
E) $- 2 x \left( 5 ^ { x ^ { 2 } } \right) + 1$

Question

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

A) $x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
B) $- x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
C) $2 x \left( 5 ^ { x ^ { 2 } } \right)$
D) $- 2 x \left( 5 ^ { x ^ { 2 } } \right)$
E) $- 2 x \left( 5 ^ { x ^ { 2 } } \right) + 1$

Quizplus · Accepted Answer

The correct answer is obtained by applying the Fundamental Theorem of Calculus part 1 and the chain rule. The derivative inside the integral with respect to $x$ gives $5^{x^2}$, and then multiplying by the derivative of the upper limit $x^2$ with respect to $x$, which is $2x$, results in $-2x(5^{x^2})$. The negative sign comes from the fact that the upper limit is $x^2$ and the lower limit is 0, so when differentiating, the sign changes.

The Derivative ddx[∫x205tdt]\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]dxd​[∫x20​5tdt]

The Derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$