How do you differentiate f(x)= (7e^x+x)^2 using the chain rule.?

1 Answer
Dec 18, 2015

f'(x) = 2(7e^x + x)(7e^x + 1)

Explanation:

With the chain rule, if you have some composition of functions that looks like

f(x) = g(h(x)),

then the derivative f'(x) is equal to

f'(x) = d/(dh)g(h)*d/(dx)h(x).

Essentially, differentiate the outside function while treating the whole inside function as if it's a single variable, and multiply it by the derivative of the inside function.

To illustrate what I mean, just imagine h = 7e^x + x for a second.

Then we have

f(x) = g(h(x)) = h^2.

Right? g(h) is just all of h squared, and so is f(x).

So, to find the derivative, we'll just apply the formula we have above.

f'(x) = d/(dh)g(h)*d/(dx)h(x)

The derivative of g(h) = h^2 with respect to h is just 2h. (power rule)

And, the derivative of h with respect to x is h' = 7e^x + 1.

Let's plug all that in:

f'(x) = 2h*h'

-> f'(x) = 2(7e^x + x)(7e^x + 1)

And there's our answer.