Let f be a differentiable function such that f(2)=2 and f′(2)=5. If g(x)=x^3f(x), what is the value of g′(2) ?

Answer 1

Let us apply the product rule of differentiation of a product of two functions. The product rule states that

d/dx (fg) = d/dx (f) * g + f * d/dx(g)

or

(fg)'(x) = f'(x) * g(x) + f(x) * g'(x).

Observe that if g(x) = x^3 * f(x) then

g'(x) = 3x^2 * f(x) + x^3 * f'(x).

Thus, g'(2) = 3 * 2^2 * f(2) + 2^3 * f'(2) = 3 * 4 * 2 + 8 * 5 = 24 + 40 = 64.