Inverse, Exponential, and Logarithmic Functions


Derivatives of e x and of the Natural Log

Derivatives of Exponential Functions

One truly remarkable characteristic of e x is that

e x = e x    

Besides the trivial case of f (x) = 0 , e x and its constant multiples are the only functions whose derivatives are equal to themselves!

Incorporating the principles of the chain rule, we might also say that if u is a function of x , then

e u = e u    

By extension, e x dx = e x + c . Using the fact that

e u = e u    

we can derive a more general formula for the derivative of a x , where a is any positive constant.

First, note that a x can be rewritten as

e ln(ax)

since e x and ln(x) are inverses of each other.

This quantity can also be written as

e x ln(a) .

Now,

a x = e x ln(a)    

By the chain rule,

a x = e x ln(a) ln(a)    

Changing the exponential function back to its original form, we get

a x = a x ln(a)    

By extension,

a x dx= a x+c    

Derivatives of Logarithms

It may be satisfying to learn now that for x >0,

ln(x) =    

The appeal rests in the corresponding implication that

=ln x +c    

Recall that the power rule did not offer a way of integrating the function , but now it is possible to do so.

A related rule for logarithms of any base is that

loga(x) =    

Logarithmic Differentiation

To find the derivative of a constant raised to a power of x , the rule presented earlier in this section should suffice. However, to find the derivative of a function of x that is raised to a power of x , the technique of logarithmic differentiation is necessary.


Example: Differentiate y = x 3x .

Step One: Take the natural log of both sides of the equation: ln(y) = ln(x 3x) .


Step Two: Now use log rules to take the variable x out of the exponent and turn it into a product: ln(y) = (3x)(ln(x)) .


Step Three: Implicitly differentiate both sides with respect to x (remember to use the chain rule):

= 3x +3 ln(x)    


Step Four: Solve for algebraically:


  = 3+3 ln(x) y  
  = 3+3 ln(x) x 3x  
  = 3x 3x +3x 3xln(x)  

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