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Mathematics LibreTexts

Implicit Differentiation

  • Page ID
    624
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    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
      \[ x^2 - y^2 = 4\]

    Larry Green (Lake Tahoe Community College)