Chapter 7.2, Problem 10E

To determine

To verify the antiderivative formula using differentiation.

Answer to Problem 10E

  ${\text{ The antiderivative of 5}}^x\text{ is }\frac{1}{\ln 5}{\text{5}}^x+C.$

Explanation of Solution

Given information: $\text{The equation is},{\displaystyle \int {\text{5}}^{x}dx=\frac{1}{\ln 5}{\text{5}}^x+C}$

Calculation:

  $\begin{array}{l}\text{To verify the antiderivative rule using diffentiation, find the derivative of}\\ {\text{the right side of the formula to show it equals 5}}^x.\\ {\displaystyle \int {\text{5}}^{x}dx=\frac{1}{\ln 5}{\text{5}}^x+C}\\ \frac{d}{dx}{\displaystyle \int {\text{5}}^{x}dx=\frac{d}{dx}\left(\frac{1}{\ln 5}{\text{5}}^x+C\right)}\left(\text{Take the derivative both sides}\right)\\ {\text{ 5}}^x=\frac{1}{\ln 5}{\text{.5}}^x\ln 5+0\left(\text{use the derivative rule }\frac{d}{dx}\left(b^x\right)=b^x\ln b\right)\\ {\text{ 5}}^x={\text{ 5}}^x\\ {\text{Therefore, the antiderivative of 5}}^x\text{ is }\frac{1}{\ln 5}{\text{5}}^x+C.\end{array}$