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At a Glance - Using Leibniz Notation

Some 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.

With these things in mind, we're ready to get cracking.

Example 1

Find  given that

x2 + y2 = 4.


Example 2

What is 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,

what is ?


Exercise 1

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

  • 4y = x

Exercise 2

In the following equation what is the derivative of y with respect to x?

  •  sin y = x2

Exercise 3

Find the derivative of y with respect to x in the following equation

  •  ln (xy) = x + y

Exercise 4

Find in the following equation.

  • y3 + x2yx = y

Exercise 5

What is in the following equation?


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