The function e and ln 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.