## Implicit Differentiation

Suppose you don't have an explicit expression for a function, but you wish to find an expression for its derivative.  It is sometimes possible to do this using implicit differentiation.  The method is so simple that it hardly deserves a name, but I will present it here because it is a useful method to keep in mind and many calculus textbooks have entire section on the topic.  Suppose you know the function

and you need an expression for $dg/dx$. One way of proceeding is to invert the equation so that you have

and then differentiate, but this might me quite difficult in some cases.  It may even be impossible to properly invert the equation. Rather than wasting time with algebra, you can do the following: Differentiate the equation to yield

then solve for $dg/dx$

That's all there is to it!

Example 1: Given

find $dy/dx$.

Solution: Differentiate the equation with respect to $x$

now solve for $dy/dx$

Example 2: See inverse trig derivatives