Question

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

Answer 1

`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.