Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).]

Question

e (raised to the exponent) 5 radical x

g(x), f(u) = ?

Find the derivate dy/dx = ?

Answer 1

y = e^5√x

Let g(x) = 5√x and f(x) = e^x. Then f'(x) = e^x and g'(x) = (5/2)x^-1/2 = 5 / (2√x).

So f(g(x)) = f(5√x) = e^5√x

d/dx[f(g(x))] = f'(g(x))g'(x) = f'(5√x)(5 / (2√x)) = e^5√x (5 / 2√x) = (5/2)(e^5√x / √x)