What is implicit differentiation?
- An equation connecting x and y is not always easy to write explicitly in the form y= f(x) or x = f(y)
- However you can still differentiate such an equation implicitly using the chain rule:
- Combining this with the product rule gives us:
- These two special cases are especially useful:
- When x and y are connected in an equation you can differentiate both sides with respect to x and rearrange to find a formula (usually in terms of x and y ) for dy/dx
- Note that dy/dx is a single algebraic object
- When rearranging do not treat dy/dx as a fraction
- Especially do not try to separate dy and dx and treat them as algebraic objects on their own!
- When using implicit differentiation you will not always be able to write dy/dx simply as a function of x.
- However, this does not stop you from answering questions involving the derivative.