* Site-Outage Notice: Our engineering elves will be tweaking the Shmoop site from Monday, December 22 10:00 PM PST to Tuesday, December 23 5:00 AM PST. The site will be unavailable during this time.
Dismiss
© 2014 Shmoop University, Inc. All rights reserved.
Computing Derivatives

Computing Derivatives

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

Therefore

7x

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 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.

Advertisement
Noodle's College Search
Advertisement
Advertisement
Advertisement