Question

The function g(x)=ln(f(x)) passes through the origin. Show why the derivatives of f(x) and g(x) are equal when x = 0. Show all your work.

Answer 1

g(x) = ln(f(x))

g'(x) = 1/f(x) f'(x)

g(0) = 0

ln(f(0)) = 0

f(0) = 1

Therefore:

g'(0) = 1/f(0) f'(0)

g'(0) = 1/1 f'(0)

g'(0) = f'(0)

Answer 2

Explanation:

g(x) = ln(f(x))

Take derivatives of both sides.

g'(x) = 1/f(x) f'(x)

g(x) passes through the origin, so g(0) = 0:

g(0) = 0

ln(f(0)) = 0

f(0) = 1

Therefore:

g'(0) = 1/f(0) f'(0)

g'(0) = 1/1 f'(0)

g'(0) = f'(0)