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3.4 Chain Rule
Prerequisite Knowledge
• Product Rule
• Quotient Rule
• Power Rule
• Special Derivatives:
Special Derivatives
€
d
dxsin x( )[ ] = cos x( )
€
d
dxcos x( )[ ] = −sin x( )
€
d
dxex
[ ] = ex
€
d
dxln x( )[ ] =
1
x
€
d
dxtan x( )[ ] = sec2 x( )
Chain Rule with Numbers
€
1
3=
1
2⋅2
3
Chain Rule With Calculus
• Why is this helpful?
• Suppose you had to differentiate:€
dy
dx=
dy
du⋅
du
dx
€
y = sin 3x 2 + 2x −1( )
Example:
€
y = sin 3x 2 + 2x −1( )
€
let u = 3x 2 + 2x −1
then du
dx= 6x + 2 and y = sin u( )
€
if y = sin u( ) then dy
du= cos u( )
€
By the Chain Rule, dy
dx=
dy
du⋅du
dx
€
So, dy
dx= 6x + 2( ) cos 3x 2 + 2x −1( )( )
Examples
€
Differentiate : y = e2x −7
€
Differentiate : y = 3x cos x 2( )
€
Determine the equation of the tangent line to
y = −2x ⋅ln x 2 + 4( ) at the origin.
Chain Rule
• Some special derivatives that come from the Chain Rule:
€
Let u be some function of x.
€
d
dxsin u( )[ ] = cos u( ) ⋅ ′ u
€
d
dxcos u( )[ ] = −sin u( ) ⋅ ′ u
€
d
dxtan u( )[ ] = sec2 u( ) ⋅ ′ u
€
d
dxln u( )[ ] =
′ u
u€
d
dxeu
[ ] = ′ u ⋅eu
Homework
• Pg. 162 # 55-92 [6]
Applications (Day 2)
• Example:
• Air is being pumped into a spherical balloon so that the radius is increasing at a rate of 2 inches per second. At what rate is the volume increasing after 3 seconds? After 10 seconds?
Example 2
• A 15 foot tall pole that was initially vertical begins to fall in such a way that the angle relative to the ground is decreasing at a rate of 3 degrees per second. At what rate is the top of the pole getting closer to the ground after 4 seconds?
Homework (2)
• Pg. 164 # 160-162
Homework (3)
• Pg. 164 # 163, 165