Derivative of a^x

The functions e^x and ln x are inverses of each other. The part we care about right now is that for any positive real number a,

e^{ln a} = a.

If we turn this equation around, we can write any positive real number a as

e^{ln a}.

For example,

7 = e^{ln 7}

Therefore

7^x

is the same thing as

(e^{ln 7})^x

which by rules of exponents is equal to

e^{(ln 7)x}.

We can find the derivative of

h(x) = e^{(ln 7)x}

using the chain rule. The outside function is

e^{□},

whose derivative is also

e^{□},

and the inside function is

(ln 7)x,

whose derivative is the constant

(ln 7).

The chain rule says

h ' (x) = e^{(ln 7)x} · (ln 7)

Turning e^{(ln 7)x} back into 7^x, we see that

h ' (x) = 7^x · (ln 7).

This is where we find the rule for taking derivatives of exponential functions that are in other bases than e.