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.