Implicit Differentiation

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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:

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

Example
Find dy/dx implicitly for the circle

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

Solution

d/dx (x² + y²) = d/dx (4)

or

2x + 2yy' = 0
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:

\[ (xy)' + \left(\dfrac{x}{y}\right)' = (5)' \]
Using the product rule and the quotient rule we have
\[ xy' + y + \dfrac{y - xy'}{ y^2} = 0 \]
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

Let
\[ 3x^2 - y^3 = 4x \text{cos}\; x + y^2 \]
Find dy/dx
Find dy/dx at (-1,1) if
\[ (x + y)^3 = x^3 + y^3 \]
Find dy/dx if
\[ x^2 + 3xy + y^2 = 1\]
Find y'' if
\[ x^2 - y^2 = 4\]