2.5 Implicit Differentiation
Implicit and Explicit Functions
Explicit Function
y
Implicit Function
xy  1
1
x
But what if you have a function like this….
x  2y  4y  2
2
3
To differentiate: Use implicit differentiation.
Implicit Differentiation
To find dy/dx implicitly, you must realize
that the differentiation is taking place with
respect to x.
x alone
y?


derive as usual
need to use chain rule!
Examples:
Differentiating with Respect to x
a.
d
dx
x 
3
Variables agree  use simple power rule
b.
d
dx
y 
3
Variables disagree  use chain rule
Examples:
Differentiating with Respect to x
c.
d
dx
d.
d
dx
x  3 y 
xy 
2
Guidelines for Implicit Differentiation
1. Differentiate both sides of the equation with
respect to x.
2. Collect all terms involving dy/dx on the left side
and all other terms to the right.
3. Factor out dy/dx.
4. Solve for dy/dx
Example 1:
x  y 1
2
2
This is not a function,
but it would still be
nice to be able to find
the slope.
Example 2:
2 y  x  sin y
2
This can’t be solved for y.
Example 3:
Find the equation of the tangent line to the curve
x  xy  y  7 at (-1, 2)
2
2
Example 4:
Higher Order Derivatives
2
Find
d y
dx
2
if 2 x 3  3 y 2  7 .
More Examples
1)
y  y  5 y  x  4
2)
x  2y  4y  2
3)
x  4y  4
4)
3 x  y
3
2
2
3
2

2
2
2

2 2

Find the slope at 

 100 xy
1 
2 ,
.
2 
Find the slope at 3 ,1 .