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Mathematics LibreTexts

Implicit Differentiation

Implicit and Explicit Functions

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

\[  y = \text{sin}\; x \]

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

\[ x^2 + y^2 = 4 \]

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

\[ x^2 + y(x)^2 = 4\]

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

\[   x^2 + y^2 = 4 \]

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

\[ xy + \dfrac{x}{y}  =  10\]

Solution: 

  1. \[ (xy)' + \left(\dfrac{x}{y}\right)' = (5)' \]
    Using the product rule and the quotient rule we have

  2. \[ xy' + y + \dfrac{y - xy'}{ y^2} =  0 \]

  3. 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    
    \[ 3x^2 - y^3  = 4x \text{cos}\; x + y^2 \]
    Find dy/dx

  2. Find dy/dx at (-1,1) if
    \[  (x + y)^3 = x^3 + y^3  \]

  3. Find dy/dx if
    \[ x^2 + 3xy + y^2 = 1\]

  4. Find y'' if
    \[ x^2 - y^2 = 4\]

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