Question

Let f be a differentiable function such that f'(x) = 2x and f(1) = 3. If g(x) = f(x²), what is the value of g'(1)?

Answer 1

To find the derivative of the function g(x) = f(x²) at x = 1, we apply the chain rule. The given information that f'(x) = 2x is used to evaluate f'(x²) and then multiplied by the derivative of the inner function, which is 2x. The final answer for g'(1) is 4.

Step-by-step explanation:

The student is asking to find the derivative of the composite function g(x) = f(x²) at x = 1, given that f'(x) = 2x and f(1) = 3. To solve this, we use the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Here's the step-by-step explanation:

1. Find g'(x) by applying the chain rule: g'(x) = f'(x²) · 2x.
2. Substitute x = 1 into f'(x²), which gives us f'(1)= 2·1 = 2.
3. Now, substitute x = 1 into 2x, which gives us 2(1) = 2.
4. Multiply the results from steps 2 and 3 to find g'(1): g'(1) = 2 · 2 = 4.

Therefore, the value of g'(1) is 4.