# 2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order

```2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2.3 Product and Quotient Rules, Trig, and Higher Order Derivatives
Objective: You will be able to:
• use the Product Rule
• use the Quotient Rule
• find the derivative of all trigonometric functions
• find a higher order derivative of a function
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Stand and Deliver
Product Rule
d [f(x)g(x)] =[f(x)g '(x) + g(x)f '(x)]
dx
"first times derivative of the second plus
second times the derivative of the first"
2.3
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Examples
1.
f(x) = (3x)(x2 + 1)
one option (distribute and use power rule)
another option (product rule)
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Examples
2.
f(x) = x cosx
no choice, must use product rule
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
y = (x2 + 2x)(x + 1)
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Stand and Deliver
Calculus the Musical: Quotient Rule
Quotient Rule
[
[ =
d f(x)
dx g(x) g(x)f '(x) ­ f(x)g'(x)
[g(x)]2
"lo D hi, minus hi D lo, over lo lo"
2.3
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Examples
1.
g(x) = 3x ­ 1
x2 + 3
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2.
y = x2 +3x 6
trick question
don't need quotient rule
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
3. Use the Quotient Rule to find d [tan x]
dx
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Stand and Deliver
Calculus the Musical: Triggy Rules
now add other 4 trig functions
2.2/2.3
Derivative of Trig. Functions
d [tan x] = sec2 x
d [sin x] = cos x
dx
dx
d [cot x] = ­ csc2 x
d [cos x] = ­ sin x
dx
dx
d [sec x] = sec x tan x
dx
d [csc x] = ­ csc x cot x
dx
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Notice Patterns:
the derivatives of the "c's" always negative
(cosine, cotangent, and cosecant­­all have negative derivatives)
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Warm Ups
find the derivative...
1. y = x tan x
2. f(x) = cos x
sin x
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
Stand and Deliver
Higher Order Derivatives
2.3
s(t) = position
v(t) = s'(t) = velocity
a(t) = v'(t) = s''(t) = acceleration speed = Iv(t)I
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
1. If the nth derivative of y is denoted as yn and y = -sin x, then y7
is the same as:
(A) y
(B) dy
dx
(C) d2y (D) d3y (E) none of the above
dx2
dx3
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
2. Use the table below to find the value of
(A) 5/2 (B) -3/2 (C) -13
x
1
3
(D) 12
f(x) g(x)
4
7
2
­ 4
at x = 3.
(E) 21/2
f ' (x) g ' (x)
5
­ 1
2.3 2011 Product Rule, Quotient Rule, Trig and Higher Order Derivatives started.notebook
September 12, 2013
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