Implicit Differentiation

Implicit and Explicit Functions

An explicit function is an function expressed as y = f(x) such as

        y = sinx

y is defined implicitly if both x and y occur on the same side of the equation such as

        x2 + y2 = 4

we can think of y as function of x and write:

        x2 + y(x)2 = 4

Implicit Differentiation

To find dy/dx, we proceed as follows:

  1. Take d/dx of both sides of the equation remembering to multiply by y' each time you see a y term.

  2. Solve for y'

Example

Find dy/dx implicitly for the circle 

        x2 + y2 = 4

Solution

  1.         d/dx (x2 + y2)  =  d/dx (4)

    or

            2x + 2yy'  =  0

  2. Solving for y, we get

                2yy'  =  -2x

                y'  =  -2x/2y

                y'  =  -x/y

Example:  

Find y' at (4,2) if 

        xy + x/y  =  10

Solution:  

  1.         (xy)' + (x/y)' = (5)'

    Using the product rule and the quotient rule we have

                            y - xy'
            xy' + y +                =  0
                                y2

  2. Now plugging in x = 4 and y = 2,

                             2 - 4y'
            4y' + 2 +                =  0
                                22
       

            16y' + 8 + 2 - 4y' = 0         Multiply both sides by 4

            12y' + 10  =  0

            12y' = -10

            y' = -5/6

 

Exercises:

  1. Let    

            3x2 - y3  = 4x cosx + y2

    Find dy/dx

  2. Find dy/dx at (-1,1) if

            (x + y)3 = x3 + y3  

  3. Find dy/dx if

            x2 + 3xy + y2 = 1

  4. Find y'' if

            x2 - y2 = 4

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