Inverse, Exponential, and Logarithmic Functions: Derivatives of e x and of the Natural Log

a^xdx=

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

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

log_a(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):