give answer in 2 step with explanation at the end of each step and final answer at the end:15. f(z) = e72*
[© Answer |

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:15. f(z) = e72*
[© Answer |

VIPSolutions · Accepted Answer

## Problem
Given:  
$ f(x) = e^{-2x^2} $

Let's find the derivative $ f'(x) $.

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### **Step 1: Identify the Rule and Set Up the Derivative**

- The function is $ f(x) = e^{u(x)} $ where $ u(x) = -2x^2 $.
- The derivative of $ e^{u(x)} $ is $ e^{u(x)} \cdot u'(x) $.

**Explanation:**  
We use the chain rule because the exponent is a function of $ x $.

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### **Step 2: Differentiate the Exponent**

- $ u(x) = -2x^2 $
- $ u'(x) = -4x $
- Now apply the chain rule:  
  $ f'(x) = e^{-2x^2} \cdot (-4x) $

**Explanation:**  
Differentiate the exponent and multiply by the original exponential function.

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## **Final Answer**

$$
f'(x) = -4x e^{-2x^2}
$$

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