If f(0) = f(1) = 0, f’(1) = 2 and y = f(e x ) e f(x), write the value of dy/dx at x = 0

Question

If f(0) = f(1) = 0, f’(1) = 2 and y = f(e^x) e^f(x), write the value of \(\cfrac{dy}{d\mathrm x} \) at x = 0.

Answer 1

 

Using the Chain Rule of Differentiation,

\(\cfrac{dy}{d\mathrm x} \) = u . v' + u' .v

= f(e^x ). e ^f(x) f ’(x) + f ’(e^x )e^x . e ^f(x)

At x = 0,

\(\cfrac{dy}{d\mathrm x} \) = f(e⁰).e^f(0)f'(0) + f'(e⁰)e⁰. e^f(0)

= f(1). e^f(0) f’(0) + f’(1). e^f(0)

= 0. e⁰ f’(0) + 2.e⁰

= 0 + 2.1

= 2