To find the derivative of the function \( e^{-2} \), we need to apply the chain rule. 

Given:
\[ y = e^{-2} \]

The exponent \(-2\) is a constant, and the derivative of an exponential function \(e^u\) with respect to \(u\) is \(e^u \cdot \frac{du}{dx}\). 

In this case, \( u = -2 \), which is a constant. Therefore, \(\frac{du}{dx} = 0\).

Applying the chain rule:
\[ \frac{dy}{dx} = e^{-2} \cdot \frac{d(-2)}{dx} \]

Since \(\frac{d(-2)}{dx} = 0\):
\[ \frac{dy}{dx} = e^{-2} \cdot 0 = 0 \]

So, the derivative of \( e^{-2} \) is:
\[ (e^{-2})' = 0 \]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the derivative of \( e^{3x} \)?
2. How do you find the derivative of a constant function?
3. What is the chain rule in differentiation?
4. Can you explain how to differentiate \( e^{x^2} \)?
5. How does the derivative of \( e^x \) change with different exponents?

**Tip:** When differentiating exponential functions, always consider if the exponent is a function of \(x\) or a constant. This will determine whether you need to use the chain rule or not.

This page provides a detailed step-by-step solution for finding the derivative of e^-2. Learn how to apply the chain rule in differentiation when the exponent is a constant. This problem involves the derivative of exponential functions and is suitable for students in Grades 11-12 or early university level calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation