Q. 44 - Calculus

Question

Combining derivatives and integrals: Simplify each of the following as much as possible: 

$\frac{d}{dx}{\int }_0^x{e}^{-t^2}dt$

Step-by-Step Solution

Answer

The derivative of given function is $e^{-x^2}.$

Step 1. Given information

The derivative is-

$\frac{d}{dx}{\int }_0^x{e}^{-t^2}dt$

Step 2. Calculation

The derivative is $\frac{d}{dx}{\int }_0^x{e}^{-t^2}dt$

Now, if $f$ is continuous on $[a,b]$ then for all $x\in [a,b]$

$$\begin{gathered} \frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right) \\ So, \\ f\left(t\right)=e^{-t^2} \\ f\left(x\right)=e^{-x^2} \end{gathered}$$

The derivative expression can be written as,

$$\begin{gathered} \frac{d}{dx}{\int }_0^x{e}^{-t^2}dt \\ =e^{-x^2} \left[f\right(x)=e^{-x^2}] \end{gathered}$$