Find h(x)?

Question

If f and g are twice differentiable functions such that g(x)=e^f(x) and g"(x)=h(x)e^f(x), find h(x).

 

a) f'(x)+f"(x)

b) f'(x)+(f"(x))^2

c) (f'(x)+f"(x))^2

d) (f'(x))^2+f"(x)

e) 2f'(x)+f"(x)

 

Answer: D

 

Please show all your work.

Answer 1

Use the chain rule and the product rule:

g(x) = e^f(x)

g'(x) = f'(x) e^f(x)

g''(x) = f''(x) e^f(x) + (f'(x))² e^f(x) = h(x) e^f(x),

 

where h(x) = f''(x) +(f'(x))² .