Find the derivative of f(x) = e^(x^2).
Correct Answer: 2x e^(x^2)
- Step 1: Identify the function you want to differentiate, which is f(x) = e^(x^2).
- Step 2: Recognize that this is a composite function, where the outer function is e^u (with u = x^2) and the inner function is u = x^2.
- Step 3: Apply the chain rule, which states that the derivative of e^u with respect to x is e^u * (du/dx).
- Step 4: Find the derivative of the inner function u = x^2. The derivative du/dx = 2x.
- Step 5: Substitute back into the chain rule formula: f'(x) = e^(x^2) * (du/dx).
- Step 6: Replace du/dx with 2x: f'(x) = e^(x^2) * 2x.
- Step 7: Rearrange the expression to get the final answer: f'(x) = 2x e^(x^2).
- Chain Rule – The chain rule is a fundamental differentiation technique used to find the derivative of composite functions.
- Exponential Functions – Understanding the properties of exponential functions, particularly how they behave under differentiation.
