Implicit Differentiation

Implicit and Explicit functions

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

        y  =  2x³ + 5

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 (2,2) if 

        xy + x/y  =  5

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  =  2 and y  =  2,
   
           2y' + 2 + (2 - 2y')/4  =  0         Multiply both sides by 4

           8y' + 8 + 2 - 2y'  =  0

           6y'  =  -10

           y'  =  -5/3
   

Exercises:

1. Let    
   
           3x² - y³   =   4x + y²
   
   Find dy/dx
   
   

2. Find dy/dx at (-1,1) if
   
           x + y  =  x³ + y³  
   
   

3. Find dy/dx if
   
           x² + 3xy + y² = 1
   
   

4. Find y'' if
   
           x² - y²  =  4

Application

Example

Suppose that the demand function for a boat shop is given by 

        p  =  -0.01x³ + x + 10,000

Find the rate of change of x with respect to p when x  =  20.  A boat craftsman can think of this question as who fast will the number of boats she will need to build change as the price is increased.  Solving for x in terms of p is nearly impossible.  Instead, we can differentiate implicitly.

        1  =  -0.03x² x' + x'

Now plug in 20 for x to get

        1  =  -0.03(20)² x' + x'

        1  =  -12x' + x'  =  -11x'

        x'  =  -1/11

The rate of change is -1/11 boats per dollar increase.

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