Implicit Differentiation (Cambridge International AS Mathematics)

Implicit differentiation

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!

Exam Hint

• 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.

Worked Example