Section Summary

We have accomplished three things in this section:

• a specific formula for the derivative of the natural logarithm function:

  ${\displaystyle \frac{d}{dx}\ln x=\frac{1}{x};}$

• a general formula for derivatives of inverse functions:

  ${\displaystyle \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}};}$

• the technique of implicit differentiation.

If `y` is defined implicitly as a function of `x` by an equation relating `y` and `x`, we can find an expression for the derivative `dytext[/]dx` (in terms of both `x` and `ytext[)]` by differentiating both sides of the equation with respect to `x` and solving the resulting equation for `dytext[/]dx.`