Why is the "inside function" the exponent itself when differentiating e^x, but not for other expressions raised to some power?

Question

Using the chain rule for functions raised to some power, like (2x+4)^2, the "inside function" is considered the base of the power. But for exponential functions like e^(2x+4) the inside function is the exponent itself.

Why is that? How do you determine which function is the inside/outside when using the chain rule?

Depends which contains $x$. If both do, you need another trick. — Wojowu, Nov 13 '16 at 12:19
The "inside function" is the function you'd evaluate first if you evaluated the composition function. Use the order of operations. — David Mitra, Nov 13 '16 at 12:23

Gnuk · Accepted Answer · 2018-08-23T18:56:09.537

3

The chain rule states that the derivative of $f(g(x))$ is equal to $f'(g(x)) \cdot g'(x)$. Here, $g$ is the inner function, and $f$ is the outer function.

If you want to write your examples in this way, then in the first case we have (for example) $f(x) = x^2$ and $g(x) = 2x+4$, as then $f(g(x)) = f(2x+4) = (2x+4)^2$. Hence $2x+4$ is the inner function.

In the second case, we have (for example) $f(x) = e^x$ and $g(x) = 2x + 4$, as then $f(g(x)) = f(2x+4) = e^{2x+4}$. Hence $2x+4$ is the inner function.

edited Aug 23 '18 at 18:56

answered Nov 13 '16 at 12:21

Gnuk

461

So the inner function is the one with the variable in the expression? What if it were x^(2x+4) – Pineapple29 Nov 13 '16 at 12:25
That expression is somewhat more difficult to get into the right form: try $x = e^{\ln x}$ to rewrite it. – Gnuk Nov 13 '16 at 13:07

Why is the "inside function" the exponent itself when differentiating e^x, but not for other expressions raised to some power?

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