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

A) $e ^ { x ^ { 2 } } - 1$
B) $e ^ { x ^ { 2 } }$
C) $\frac { e ^ { x ^ { 2 } } } { 2 x }$
D) $2 x e ^ { x ^ { 2 } }$
E) $2 x e ^ { x ^ { 2 } } - 1$

Question

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

A) $e ^ { x ^ { 2 } } - 1$
B) $e ^ { x ^ { 2 } }$
C) $\frac { e ^ { x ^ { 2 } } } { 2 x }$
D) $2 x e ^ { x ^ { 2 } }$
E) $2 x e ^ { x ^ { 2 } } - 1$

Quizplus · Accepted Answer

The derivative of the integral from a constant to x of a function is just the function evaluated at x, according to the Fundamental Theorem of Calculus. Therefore, the derivative of $\int _ { 0 } ^ { x } e ^ { t ^ { 2 } } d t$ is $e^{x^2}$.

The Derivative ddx[∫0xet2dt]\frac { d } { d x } \left[ \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } d t \right]dxd​[∫0x​et2dt]

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