Chapter 4.4, Problem 58E

To determine

To calculate: The derivative of y with respect to x, for $y=e^{2x}\text{is }2(\ln 2)e^{2x}.$

Answer to Problem 58E

The derivative is $\frac{dy}{dx}=e^{2x}\times 2$ ,So given statement is false.

Explanation of Solution

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

  $y=e^{2x}$

Formula Used:

  $\text{Chain Rule:}\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$

Calculation: Using the differentiation,

  $\begin{array}{l}y=e^{2x}\\ \text{let us assume u=2x}.\\ \text{Differentiating u with respect to x,}\\ \frac{du}{dx}=2\\ \text{Now, differentiating y with respect to u},\\ \frac{dy}{du}=\frac{d{e}^u}{du}\\ \frac{dy}{du}=e^u\\ \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\\ \frac{dy}{dx}=e^{2x}\times 2\end{array}$

So, the derivative is $\frac{dy}{dx}=e^{2x}\times 2$ and statement is false.