Chain Rule

Idea. Use the chain rule: derivative of is . The outer constant stays outside.
Step 1. Let .
Step 2.

Differentiate termwise: ; by chain rule. Add:

Chain rule: with . So derivative .

We are asked to differentiate , or . Step 1: Apply the power rule
The power rule for differentiation is given by: We can treat as . Differentiating using the power rule: This simplifies to: Thus, the derivative of is .

We are asked to differentiate . Step 1: Apply the chain rule
Since is in the exponent, we apply the chain rule:
Step 2: Differentiate the exponent
The derivative of is , so: Thus, the derivative of is .

We are asked to differentiate: Use the chain rule for differentiation. First, differentiate the outer function: where . Next, differentiate the inner function : Thus, the final derivative is: Thus, the correct answer is .