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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

eln a.

For example,

7 = eln 7

Therefore

7x

is the same thing as

(eln 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 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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