Let's solve the given problem step by step.

Given:

$y = -4e^{-x}$

We need to calculate $\frac{dy}{dx}$.

Step 1: Differentiate $y$ with respect to $x$

Using the chain rule: $\frac{d}{dx} \left[ -4e^{-x} \right] = -4 \cdot \frac{d}{dx} \left( e^{-x} \right)$

The derivative of $e^{-x}$ is $-e^{-x}$, so: $\frac{dy}{dx} = -4 \cdot (-e^{-x}) = 4e^{-x}$

Step 2: Simplify

Since $e^{-x}$ is the reciprocal of $e^x$, we can rewrite the result as: $\frac{dy}{dx} = \frac{4}{e^x}$

Step 3: Match the options

The correct answer is: a. $\frac{4}{e^x}$

Let me know if you'd like further clarification! 😊

Follow-up Questions:

1. What is the chain rule and when do we use it in differentiation?
2. How does the exponential function $e^x$ behave as $x \to \infty$ and $x \to -\infty$?
3. Why is the derivative of $e^{-x}$ equal to $-e^{-x}$?
4. What does the derivative of $y = -4e^{-x}$ represent in a graph?
5. How would the derivative change if $y = -4e^{x}$?

Tip:

Always simplify exponential terms like $e^{-x}$ into fraction form $1/e^x$ when comparing options!