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axdx=ax+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

=lnx+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 = x3x.

Step One: Take the natural log of both sides of the equation: ln(y) = ln(x3x).


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)x3x  
 = 3x3x +3x3xln(x)  

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