Assume that y = f(x) is differentiable for all x

Question

Assume that y = f(x) is differentiable for all x, f(2) = 2 and f'(2) = 1. Let g(x) = f(x) ln x.

(i) Find g'(2) from the definition (first principle formula).

(ii) Let G(x) = g(f(x)), find G' (2).

Answer 1

Given:

f(2) = 2

f'(2) = 1

g(x) = f(x)•ln(x)

Derive g(x), using the Product Rule:

g'(x) = f'(x)•ln(x) + f(x)/x

(i)

If x = 2, then:

g'(2) = f'(2)•ln(2) + f(2)/2

g'(2) = f'(2)•ln(2) + f(2)/2

Substitute:

g'(2) = 1•ln(2) + 2/2

g'(2) = ln(2) + 1

(ii)

G(x) = g(f(x))

Using the Chain Rule:

G'(x) = g'(f(x))•f'(x)

If x = 2, then:

G'(2) = g'(f(2))•f'(2)

Substitute:

G'(2) = g'(2)•1

G'(2) = ln(2) + 1