What is the derivative of f(x) = sin(x^2)?
Correct Answer: 2x cos(x^2)
- Step 1: Identify the function we want to differentiate, which is f(x) = sin(x^2).
- Step 2: Recognize that this is a composite function, where the outer function is sin(u) and the inner function is u = x^2.
- Step 3: Apply the chain rule, which states that the derivative of sin(u) is cos(u) multiplied by the derivative of u.
- Step 4: Find the derivative of the inner function u = x^2. The derivative is u' = 2x.
- Step 5: Now, apply the chain rule: f'(x) = cos(x^2) * (derivative of x^2).
- Step 6: Substitute the derivative of the inner function: f'(x) = cos(x^2) * 2x.
- Step 7: Simplify the expression to get the final answer: f'(x) = 2x cos(x^2).
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