Chapter 4, Problem 34RE

To determine

To calculate: To find the values of x for which the function is differentiable.

Answer to Problem 34RE

The function is differentiable at values of x except $x=0$.

Explanation of Solution

Given Information: The given function is $y=\frac{1}{1-e^x}$ .

Calculation:

The derivative of the function $y=\frac{1}{1-e^x}$ is,

  $\begin{array}{l}y=\frac{1}{1-e^x}\\ \frac{dy}{dx}=\frac{d}{dx}\left(\frac{1}{1-e^x}\right)\\ =\frac{\left(1-e^x\right)\frac{d}{dx}\left(1\right)-1\frac{d}{dx}\left(1-e^x\right)}{{\left(1-e^x\right)}^2}\left(\text{Apply quotient rule}\right)\\ =\frac{\left(1-e^x\right)\times 0-\left(-e^x\right)}{{\left(1-e^x\right)}^2}\left(\because \frac{d}{dx}\left(\text{constant}\right)=0,\frac{d}{dx}{e}^{nx}=n{e}^{nx}\right)\\ =\frac{e^x}{{\left(1-e^x\right)}^2}\end{array}$

The function is not differentiable for all those values of x where the denominator of the derivative becomes zero.

That is,

  $\begin{array}{c}{\left(1-e^x\right)}^2=0\\ 1-e^x=0\\ 1=e^x\\ x=0\end{array}$

Hence, the function is not differentiable at $x=0$ .

The function is differentiable at values of x except $x=0$.