AP Calc Notes: MD – 4 Implicit Differentiation
The Problem
We know:
d 3
x ) = 3x2
(
dx
We want:
d 3
(y )=?
dx
d 3
y ) = 3y2
(
dy
Review
If y = g(x), then y', g',
However, x' =
Chain Rule:
Ex: a.
dy
dg
and
all mean the same thing.
dx
dx
dx
=1
dx
d
f ( g ( x ) ) = f’(g(x))g’(x)
dx
d 3
(x ) =
dx
b.
3
d
g ( x )) =
(
dx
c.
d 3
(y )=
dx
Implicit Differentiation
To evaluate
∴
d
f ( y ) , we assume y is some function of x: y = g(x)
dx
d
f ( y) =
dx
Don’t forget the baby dy
dx
Ex:
a.
d
y=
dx
b.
d
sin 2 y =
dx
c.
d
( xy ) =
dx
d.
d  y
 =
dx  x 
Ex: Find
dy
if x3 + y 3 = 1
dx
a. Using implicit differentiation.
b. By solving for y
Sometimes you need to factor to solve for
Ex: Find
dy
for x3 + 4 y 3 + 6 y 2 = 15
dx
dy
dx
Recall:
Horizontal tangent line
(slope = 0)
A fraction
a
= 0 when a = 0 and b does not.
b
To find when
and solve.
dy
= 0 , set the numerator = 0
dx
Vertical tangent line
(slope is undefined)
a
A fraction is undefined when b = 0 and
b
a does not.
dy
is undefined, set the
dx
denominator = 0 and solve.
To find when
Caution: these methods may produce extraneous roots. You must check all values of x and y in the original
equation to be sure they exist on the original curve.
a. Find the equation of any horizontal tangent lines to the curve x3 + 4 y 3 + 6 y 2 = 15
b. Find the equation of any vertical tangent lines to the curve x3 + 4 y 3 + 6 y 2 = 15