Our problem for today is to differentiate *a ^{x}* with the (usual) restrictions that

*a*is a positive number and not equal to 1. The reasoning here is very different from that for finding other derivatives and therefore I hope you and your students find it interesting.

The definition of derivative followed by a little algebra gives tells us that

.

Since the limit in the expression above is a number, we observe that the derivative of *a ^{x}* is proportional to

*a*. And also, each value of

^{x}*a*gives a different constant. For example if

*a*= 5 then the limit is approximately 1.609438, and so .

I determined this by producing a table of values for the expression in the limit near *x* = 0. You can do the same using a good calculator, computer, or a spreadsheet.

h-0.00000030 1.60943752

-0.00000020 1.60943765

-0.00000010 1.60943778

0.00000000 undefined

0.00000010 1.60943804

0.00000020 1.60943817

0.00000030 1.60943830

That’s kind of messy and would require us to find this limit for whatever value of *a* we were using. It turns out that by finding the value of *a* for which the limit is 1 we can fix this problem. Your students can do this for themselves by changing the value of *a* in their table until they get the number that gives a limit of 1.

Okay, that’s going to take a while, but may be challenging. The answer turns out to be close to 2.718281828459045…. Below is the table for this number.

h-0.00000030 0.99999985

-0.00000020 0.99999990

-0.00000010 0.99999995

0.00000000 undefined

0.00000010 1.00000005

0.00000020 1.00000010

0.00000030 1.00000015

Okay, I cheated. The number is, of course, *e*. Thus,

.

The function *e ^{x}* is its own derivative!

And from this we can find the derivatives of all the other exponential functions. First, we define a new function (well maybe not so new) which is the inverse of the function *e ^{x}* called ln(

*x*), the natural logarithm of

*x.*(For more on this see Logarithms.) Then

*a*=

*e*

^{ln(a)}and

*a*= (

^{x}*e*

^{ln(a)})

*=*

^{x}*e*

^{(ln(a)x)}. Then using the Chain Rule the derivative is

Finally, going back to the first table above where *a* = 5, we find that the limit we found there 1.609438 = ln(5).

For a video on this topic click here.

Revised 8-28-2018, 6-2-2019

Thanks. It’s a nice post about derivative of exponential function. I really like it :). It’s really helpful. Good job.

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