The derivative of f(g(x)) with respect to x is?

• A. f'(g(x)) · g'(x)
• B. f(g'(x)) · g(x)
• C. f'(x) · g'(x)
• D. f'(x) + g'(x)

Answer: (A)

Chain rule for a composition: when you differentiate f(g(x)), you differentiate the outer function with respect to its input and multiply by the derivative of the inner function. If you set u = g(x), then d/dx f(u) = f'(u) · du/dx, and substituting back gives f'(g(x)) · g'(x).

For example, if f(u) = u^2 and g(x) = sin x, then f(g(x)) = sin^2 x and the derivative is 2·sin x · cos x, which matches f'(g(x)) · g'(x) since f'(u) = 2u and g'(x) = cos x.

So the derivative is f'(g(x)) · g'(x).