MAT200
Franklin College
Erich Prisner

Differentiating exponential and logarithmic functions

What we need in both cases is the fact that

(see here for a proof of this fact).

Derivative of ln(x)

Look at the differential quotient (without the limit, for the moment). We use the laws of logarithms and use the abbreviation  n = x/h to reformulate this into. 

Now note that  n  goes to infinity if  h  goes to 0 (feor every fixed  x). We get

and since the function  g(x) = (1+1/x)^x  is continuous, and therefore also the function  ln(g(x)), we can go on as 

Derivative of e^x

can be done in the same way ... how exactly?

Using the inverse theorem

We need only one of the two proofs above when we can use the inverse theorem for derivatives: Name  f(x)=e^x  and g(x)=ln(x). Then since  f(x)=g^-1(x)   and   g(x)=f^-1(x)

Differentiating other exponential and logarithmic functions

Since  

from the constant multiple rule there follows

and in the same way for the exponential functions, since

using the chain rule, we get also

Erich Prisner, October 2003