Derivative of ax
The functions ex and ln x are inverses of each other. The part we care about right now is that for any positive real number a,
eln a = a.
If we turn this equation around, we can write any positive real number a as
7 = eln 7
is the same thing as
which by rules of exponents is equal to
We can find the derivative of
h(x) = e(ln 7)x
using the chain rule. The outside function is
whose derivative is also
and the inside function is
whose derivative is the constant
The chain rule says
h ' (x) = e(ln 7)x · (ln 7)
Turning e(ln 7)x back into 7x, we see that
h ' (x) = 7x · (ln 7).
This is where we find the rule for taking derivatives of exponential functions that are in other bases than e.