Chapter 4.4, Problem 27E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 27E

The derivative is $\frac{dy}{dx}=\frac{1}{e^x\times \ln 10}$ .

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y={\log }_{10}{e}^x$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}y={\log }_{10}{e}^x\\ y=\frac{\ln \left(e^x\right)}{\ln 10}\\ \frac{dy}{dx}=\frac{d\left(\frac{\ln e^x}{\ln 10}\right)}{dx}\\ \frac{dy}{dx}=\frac{1}{e^x\times \ln 10}\end{array}$

So the derivative is $\frac{dy}{dx}=\frac{1}{e^x\times \ln 10}$ .