Download Section 2.4 Product and Quotient Rules

Section 2.4 Product and Quotient Rules
1. Product Rule.
If f and g are both differentiable:
d
[ f xg x ] = f x d g x d f x g x
dx
dx
dx
Alternatively, fg' = fg ' gf '
What I say to myself: 1st times the derivative of the 2nd + 2nd times the derivative of the 1st.
2. Quotient Rule
If f and g are both differentiable:
d
d
g x f x f x g x d f x dx
dx
=
dx g x
[ g x]2
Alternatively,
/
f
gf ' fg '
=
2
g
g
Common Mnemonic: “( down d up – up d down) / down down”
What I say to myself: denominator times derivative of the numerator – numerator times the
derivative of the denominator by the denominator squared.
3. All of the trigonometric functions
d
sin x = cos x
dx
d
cos x = sin x
dx
d
tan x = sec2 x
dx
d
csc x = cscxcot x
dx
d
sec x = sec xtan x
dx
d
cot x = csc2 x
dx
Common Memory Tool: derivative of something starting with “c” is negative.
Examples:
1.
f x = x sin x (Product Rule)
f ' x = x d sin x sin x d x
dx
dx
1
1
= x cos x sin x x 2
2
sin x
= xcos x 2 x
2.
2
3
5
2
5
2
2
2
1
Y u = u u u 2u One way:
2
3
3
5
2
Y u = u u u 2u u u u 2u 3
Y u = u 2 u 2 u
2
Y ' u = 3u 2u 2u
2
Another way:
3
2
5
2
Y u = u u u
2u 1 st
(product rule)
2 nd
Y ' u = u2 u3 d 5
d
u 2u2 u5 2u 2 u2u3
du
du
= u02 u3 5u 4 4u u 5 2u 22u3 3u4 2
2
4
3
3
4
3
5
5
4
= u 5u u 4u u 5u u 4u u 2u u 3u 2u 22u3 2u 2 3u 4 1
2
= 5u 4u 5u 4u
2
2
1
2
2u 3u 4u 6u
= 3u2 2u 2u 2
And, we have the same thing!
3. Show
d
tan x = sec2 x
dx
d
d sin x
tan x =
dx
dx cos x
cos x
=
d
d
sin x sin x cos x dx
dx
2
cos x
=
cos xcos x sin xsin x cos2 x sin2 x
=
Apply Pythagorean Identity
cos 2 x
cos2 x
=
1
= sec2 x
x
cos
f x =
4. Find f ' (x) when
x
x
c
x
Key is to simplify first:
x 2c
f ' x =
d 2
d
x x 2 x 2c dx
dx
2
2
x c
2
=
x
x
2
2
1
x c
x
f x = 2
= 2
x
x c
x c
2
x
x c
2
x c2x x 2x 2x 32xc2x 3
2cx
=
= 2
2
2
2
2
2
x c
x c
x c
5. Find equation of a tangent line to the curve
x1
y ' x =
y=
x when x = 4.
x1
d
d
x x x1
dx
dx
2
x1
1
1 x1
x1 x 2 x1
x
2
2x
=
=
2
2
x1
x1
Find the slope at x = 4:
41
5
5 8
8
4
2
24
22
4 4
4
8
y ' 4 =
=
=
=
=
= 0.08 Now we have a slope
2
25
25
25
100
41
I need a point:
4 = 2 = 0.4
y 4 =
41 5
Find the equation:
y y 1 = m x x1 y0.4 = 0.08 x4
(4, 0.4)