# 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*.