Question

Let \( f(x) = e^{-|x|} \), where \( x \) is real. The value of \( \dfrac{df}{dx} \text{ at } x = -1 \text{ is} \)

• A. \(-e\)
• B. \( e \)
• C. \( \frac{1}{e} \)
• D. \( -\frac{1}{e} \)

Hint:

When differentiating a function involving \( |x| \), split the function into cases based on the value of \( x \).

Answer 1

The Correct Option is C

We are given the function \( f(x) = e^{-|x|} \), and we are asked to find the derivative at \( x = -1 \). 
 

Step 1: Understand the function \( f(x) \). 
Since \( f(x) = e^{-|x|} \), we need to deal with the absolute value function. For \( x = -1 \), we have \( |x| = 1 \). Therefore, for \( x = -1 \), \[ f(x) = e^{-|x|} = e^{-1}. \]

Step 2: Differentiate the function. 
The derivative of \( f(x) \) is piecewise because of the absolute value function. For \( x < 0 \), \( |x| = -x \), so: \[ f'(x) = \frac{d}{dx} \left( e^x \right) = e^x. \] Thus, at \( x = -1 \), we have: \[ f'(-1) = e^{-1} = \frac{1}{e}. \]