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
\[ y = \text{sin}\; x \nonumber \]
y is defined implicitly if both x and y occur on the same side of the equation such as
\[ x^2 + y^2 = 4 \nonumber \]
we can think of y as function of x and write:
\[ x^2 + y(x)^2 = 4 \nonumber \]
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
\[ x^2 + y^2 = 4 \nonumber \]
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
\[ xy + \dfrac{x}{y} = 10 \nonumber \]
Solution:
-
\[ (xy)' + \left(\dfrac{x}{y}\right)' = (5)' \nonumber \]
Using the product rule and the quotient rule we have -
\[ xy' + y + \dfrac{y - xy'}{ y^2} = 0 \nonumber \]
-
Now plugging in x = 4 and y = 2,
2 - 4y'
4y' + 2 +
2216y' + 8 + 2 - 4y' = 0 Multiply both sides by 4
12y' + 10 = 0
12y' = -10
y' = -5/6
Exercises
-
Let
\[ 3x^2 - y^3 = 4x \text{cos}\; x + y^2 \nonumber \]
Find dy/dx -
Find dy/dx at (-1,1) if
\[ (x + y)^3 = x^3 + y^3 \nonumber \] -
Find dy/dx if
\[ x^2 + 3xy + y^2 = 1 \nonumber \] -
Find y'' if
\[ x^2 - y^2 = 4 \nonumber \]