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Derivative of ax

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 = eln 7



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


whose derivative is also


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 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.

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