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Colorado National MonumentGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003
AP Calculus AB/BC
3.3 Differentiation Rules, p. 116
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
0dc
dx Example 1: 3y
0y
The derivative of a constant is zero.
Constant Rule, p. 116
2 2 .We saw that if , then y x y x
This is part of a pattern.
1n ndx nx
dx
Examples 2a & 2b:
4f x x
34f x x
8y x
78y x
Power Rule, p. 116
The Power Rule works for both positive and negative powers
d ducu c
dx dx Example 3
1n ndcx cnx
dx
Constant Multiple Rule, p. 117
57dx
dx
where c is a constant and u is a differentiable function of x.
5 17 5x 435x
(Each term is treated separately)
Sum and Difference Rules p. 117
d du dvu v
dx dx dx d du dv
u vdx dx dx
4 12y x x 34 12y x
where c is a constant and u is a differentiable function of x.
where u and v are differentiable functions of x.
Constant Multiple Rule, p. 117
d ducu c
dx dx
Example 4a Example 4b
34 4dy
x xdx
4 22 2y x x
Example 5Find the horizontal tangents of: 4 22 2y x x
34 4dy
x xdx
Horizontal tangents occur when slope = zero.34 4 0x x
3 0x x
2 1 0x x
1 1 0x x x
0, 1, 1x
Plugging the x values into the original equation, we get:
2, 1, 1y y y
(The function is even, so we only get two horizontal tangents.)
Example 5 (cont.)
4 22 2y x x
Example 5 (cont.)
4 22 2y x x
2y
Example 5 (cont.)
4 22 2y x x
2y
1y
Example 5 (cont.)
4 22 2y x x
Example 5 (cont.)
4 22 2y x x
First derivative (slope) is zero at:
0, 1, 1x
34 4dy
x xdx
Example 5 (cont.)
Product Rule, p. 119
d dv duuv u v
dx dx dx Notice that this is not just the
product of two derivatives.
This is sometimes memorized as: duv uv vu
dx
2 33 2 5d
x x xdx
5 3 32 5 6 15d
x x x xdx
5 32 11 15d
x x xdx
4 210 33 15 x x
2 3x 26 5x 32 5x x 2x
4 2 2 4 26 5 18 15 4 10x x x x x
4 210 33 15 x x
Example 6:
Quotient Rule, p. 120
2
du dvv ud u dx dx
dx v v
or 2
u vu uv
dv v
3
2
2 5
3
d x x
dx x
2 2 3
22
3 6 5 2 5 2
3
x x x x x
x
Example 7:
v u′ v′u
v2
Example 8
Find the tangent’s to Newton’s Serpentine at the origin and at the point (1, 2). 2
4
1
x
yx
2
22
1 4 4 2
1
x x xy
x
2 2
22
4 4 8
1
x xy
x
2
22
4 4
1
xy
x
Example 8 (cont.)
2
22
4 4
1
xy
x
Next, find the slopes at the given points.
For the point (0, 0), plug in 0 for x.y′ will be the slope.
2
22
4 0 4
0 1
y
So, m = 4 and the equation for the tangent is y = 4x since the y-intercept is 0.
For the point (1, 2), plug in 1 for x.
2
22
4 1 4
1 1
y
0y
So, m = 0 and the equation for the tangent is y = 2 since the m is 0.
Example 9: Find the first four derivatives of
3 25 2 y x x23 10 y x x
6 10 y x
6y(4) 0y
Higher Order Derivatives, p.122
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.