Higher Unit 3

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Higher Unit 3
Differentiation The Chain Rule
Further Differentiation Trig Functions
Further Integration
Integrating Trig Functions
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The Chain Rule for Differentiating
To differentiate composite functions
(such as functions with brackets in
them) we can use
Example
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The Chain Rule for Differentiating
You have 1 minute to come up with the rule.
1. Differentiate outside the bracket. 2. Keep the
bracket the same. 3. Differentiate inside the
bracket.
Good News ! There is an easier way.
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1. Differentiate outside the bracket. 2. Keep the
bracket the same. 3. Differentiate inside the
bracket.
The Chain Rule for Differentiating
Example
You are expected to do the chain rule all at once
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1. Differentiate outside the bracket. 2. Keep the
bracket the same. 3. Differentiate inside the
bracket.
The Chain Rule for Differentiating
Example
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The Chain Rule for Differentiating
Example
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The Chain Rule for Differentiating Functions
Example
The slope of the tangent is given by the
derivative of the equation.
Re-arrange
Use the chain rule
Where x 3
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The Chain Rule for Differentiating Functions
Remember y - b m(x a)
Is the required equation
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The Chain Rule for Differentiating Functions
Example
In a small factory the cost, C, in pounds of
assembling x components in a month is given by
Calculate the minimum cost of production in any
month, and the corresponding number of components
that are required to be assembled.
Re-arrange
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The Chain Rule for Differentiating Functions
Using chain rule
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The Chain Rule for Differentiating Functions
Is x 5 a minimum in the (complicated) graph?
Is this a minimum?
For x lt 5 we have (ve)(ve)(-ve) (-ve)
For x 5 we have (ve)(ve)(0) 0
x 5
For x gt 5 we have (ve)(ve)(ve) (ve)
Therefore x 5 is a minimum
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The Chain Rule for Differentiating Functions
The cost of production
Expensive components?
Aeroplane parts maybe ?
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Calculus

Revision
Differentiate
Chain rule
Simplify
Back
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Quit
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Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
15
Calculus

Revision
Differentiate
Chain Rule
Back
Next
Quit
16
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
17
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
18
Calculus

Revision
Differentiate
Straight line form
Chain Rule
Simplify
Back
Next
Quit
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Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
20
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
21
Calculus

Revision
Differentiate
Straight line form
Chain Rule
Simplify
Back
Next
Quit
22
Calculus

Revision
Differentiate
Straight line form
Chain Rule
Simplify
Back
Next
Quit
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Trig Function Differentiation
The Derivatives of sin x cos x
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Trig Function Differentiation
Example
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Trig Function Differentiation
Example
Simplify expression - where possible
Restore the original form of expression
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1. Differentiate outside the bracket. 2. Keep the
bracket the same. 3. Differentiate inside the
bracket.
The Chain Rule for DifferentiatingTrig Functions
Worked Example
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The Chain Rule for DifferentiatingTrig Functions
Example
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The Chain Rule for DifferentiatingTrig Functions
Example
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Calculus

Revision
Differentiate
Back
Next
Quit
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Calculus

Revision
Differentiate
Back
Next
Quit
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Calculus

Revision
Differentiate
Back
Next
Quit
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Calculus

Revision
Differentiate
Back
Next
Quit
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Calculus

Revision
Differentiate
Straight line form
Chain Rule
Simplify
Back
Next
Quit
34
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
35
Calculus

Revision
Differentiate
Straight line form
Chain Rule
Simplify
Back
Next
Quit
36
Calculus

Revision
Differentiate
Back
Next
Quit
37
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
38
Calculus

Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit
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You have 1 minute to come up with the rule.
Integrating Composite Functions
Harder integration
we get
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1. Add one to the power. 2. Divide by new
power. 3. Compensate for bracket.
Integrating Composite Functions
Example
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1. Add one to the power. 2. Divide by new
power. 3. Compensate for bracket.
Integrating Composite Functions
Example
You are expected to do the integration rule all
at once
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Integrating Composite Functions
Example
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Integrating Composite Functions
Example
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1. Add one to the power. 2. Divide by new
power. 3. Compensate for bracket.
Integrating Functions
Example
Integrating
So we have
Giving
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Calculus

Revision
Integrate
Standard Integral (from Chain Rule)
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Quit
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Calculus

Revision
Integrate
Straight line form
Back
Next
Quit
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Calculus

Revision
Use standard Integral (from chain rule)
Find
Back
Next
Quit
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Calculus

Revision
Integrate
Straight line form
Back
Next
Quit
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Calculus

Revision
Use standard Integral (from chain rule)
Find
Back
Next
Quit
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Calculus

Revision
Use standard Integral (from chain rule)
Evaluate
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Next
Quit
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Calculus

Revision
Evaluate
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Calculus

Revision
Find p, given
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Calculus

Revision
passes through the point (1, 2).
A curve for which
Express y in terms of x.
Use the point
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Calculus

Revision
Given the acceleration a is
If it starts at rest, find an expression for the
velocity v where
Starts at rest, so v 0, when t 0
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Integrating Trig Functions
Integration is opposite of differentiation
Worked Example
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1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.

Integrating Trig Functions
Special Trigonometry Integrals are
Worked Example
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1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.

Integrating Trig Functions
Example
Integrate
Break up into two easier integrals
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1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.

Integrating Trig Functions
Example
Integrate
Re-arrange
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Integrating Trig Functions (Area)
Example
The diagram shows the graphs of y -sin x and y
cos x

1. Find the coordinates of A
2. Hence find the shaded area

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Integrating Trig Functions (Area)
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Integrating Trig Functions
Example
Remember cos(x y)
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Integrating Trig Functions
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Calculus

Revision
Find
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Calculus

Revision
Find
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Calculus

Revision
Find
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Calculus

Revision
Integrate
Integrate term by term
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Calculus

Revision
Find
Integrate term by term
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Calculus

Revision
Find
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Calculus

Revision
passes through the point
The curve
Find f(x)
use the given point
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Calculus

Revision
passes through the point
express y in terms of x.
If
Use the point
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Next
Quit
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Calculus

Revision
passes through the point
A curve for which
Find y in terms of x.
Use the point
Back
Next
Quit
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