Introduction to :

Things to remember for implicit differentiation:

  • since y is a function of x, any derivative involving y must use the chain rule
      
  • since y is a function of x, taking the derivative of xy (or of any other product involving both x and y) requires the product rule
      
  • since y is a function of x, taking the derivative of  (or any other quotient involving both x and y) requires the quotient rule
      

Practice:

Example 1

Find  given that

x2 + y2 = 4.


Example 2

Find the derivative of y with respect to x given that

4y2 + 8y = 2x2.


Example 3

Assuming y is a function of x and

xy + x = y,

find  .


Exercise 1

Use implicit differentiation to find , assuming in the case that y is a function of x.

  • 4y = x

Exercise 2
  •  sin y = x2

Exercise 3
  •  ln (xy) = x + y

Exercise 4
  • y3 + x2y - x = y

Exercise 5


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