AQA A Level Maths: Pure复习笔记7.5.1 Implicit Differentiation

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:

7.5.1-Implicit-Diff-Illustr-1_form

7.5.1-Implicit-Diff-Illustr-1_ex

  • Combining this with the product rule gives us:

7.5.1-Implicit-Diff-Illustr-2_form

7.5.1-Implicit-Diff-Illustr-2_ex

  • These two special cases are especially useful:

7.5.1-Implicit-Diff-Illustr-3

  • 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!

7.5.1-Implicit-Diff-Illustr-4

Exam Tip

  • 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

7.5.1-Implicit-Diff-Example

 

 

 

 

 

 

转载自savemyexams

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