Chapter 7.2, Problem 9E

To determine

To verify the antiderivative formula using differentiation.

Answer to Problem 9E

  $\text{The antiderivative of }{e}^{2x}\text{ is }\frac{1}{2}{e}^{2x}+C$

Explanation of Solution

Given information: $\text{The equation is},{\displaystyle \int e^{2x}dx=\frac{1}{2}{e}^{2x}+C}.$

Calculation:

  $\begin{array}{l}\text{To verify the antiderivative rule using diffentiation, find the derivative of}\\ \text{the right side of the formula to show it equals }{e}^{2x}.\\ {\displaystyle \int e^{2x}dx=\frac{1}{2}{e}^{2x}+C}\\ \frac{d}{dx}{\displaystyle \int e^{2x}dx=\frac{d}{dx}\left(\frac{1}{2}{e}^{2x}+C\right)}\left(\text{Take the derivative both side}\right)\\ e^{2x}=\frac{1}{2}{e}^{2x}.2+0\left(\text{use the derivative rule }\frac{d}{dx}\left(e^{2x}\right)=\frac{1}{2}{e}^{2x}\right)\\ e^{2x}=e^{2x}\\ \text{Therefore, The antiderivative of }{e}^{2x}\text{ is }\frac{1}{2}{e}^{2x}+C\end{array}$