How do you differentiate f(x)=sqrt(e^(5x^2+x+3) using the chain rule?

1 Answer
Jul 10, 2016

This will require the application of the chain rule--twice.

First, for e^(5x^2 + x + 3)

Let y = e^u, and u = 5x^2 + x + 3

dy/dx = dy/(du) xx (du)/dx

The derivative of e^u is e^u. The derivative of 5x^2 + x + 3 is 10x + 1.

Hence,

dy/dx = e^u xx 10x + 1

dy/dx = (10x + 1)e^(5x^2 + x + 3)

Now for the second application of the chain rule.

Let y = sqrt(u) = u^(1/2) and u = e^(5x^2 + x + 3)

We already know the derivative of u. The derivative of y, by the power rule, is 1/2u^(-1/2) = 1/(2u^(1/2)).

Hence,

dy/dx = 1/(2u^(1/2)) xx (10x + 1)e^(5x^2 + x + 3)

dy/dx = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))

f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2sqrt(e^(5x^2 + x + 3))

f'(x) = ((10x + 1)e^(5x^2 + x + 3))/(2(e^(5x^2 + x + 3))^(1/2)

By the quotient rule of exponents: a^n/a^m = a^(n - m):

f'(x) = ((10x + 1)sqrt(e^(5x^2 + x + 3)))/2

Hopefully this helps!