Higher Unit 3 Differentiation The Chain Rule

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Higher Unit 3 Differentiation The Chain Rule
Further Differentiation Trig Functions Further Integration Integrating Trig Functions

The Chain Rule for Differentiating
To differentiate composite functions (such as functions with brackets in them) we can use: Example

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.

The Chain Rule for Differentiating
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

The Chain Rule for Differentiating
1. Differentiate outside the bracket. 2. Keep the bracket the same. 3. Differentiate inside the bracket. The Chain Rule for Differentiating Example

The Chain Rule for Differentiating
Example

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:

The Chain Rule for Differentiating Functions
Remember y - b = m(x – a) Is the required equation

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

The Chain Rule for Differentiating Functions
Using chain rule

The Chain Rule for Differentiating Functions
Is x = 5 a minimum in the (complicated) graph? Is this a minimum? For x < 5 we have (+ve)(+ve)(-ve) = (-ve) For x = 5 we have (+ve)(+ve)(0) = 0 x = 5 For x > 5 we have (+ve)(+ve)(+ve) = (+ve) Therefore x = 5 is a minimum

The Chain Rule for Differentiating Functions
The cost of production: Expensive components? Aeroplane parts maybe ?

Calculus Revision Differentiate Chain rule Simplify Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

Calculus Revision Differentiate Chain Rule Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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

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

Trig Function Differentiation
The Derivatives of sin x & cos x

Trig Function Differentiation
Example

Trig Function Differentiation
Example Simplify expression - where possible Restore the original form of expression

The Chain Rule for Differentiating Trig Functions
1. Differentiate outside the bracket. 2. Keep the bracket the same. 3. Differentiate inside the bracket. The Chain Rule for Differentiating Trig Functions Worked Example:

The Chain Rule for Differentiating Trig Functions
Example

The Chain Rule for Differentiating Trig Functions
Example

Calculus Revision Differentiate Back Next Quit

Calculus Revision Differentiate Back Next Quit

Calculus Revision Differentiate Back Next Quit

Calculus Revision Differentiate Back Next Quit

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

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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

Calculus Revision Differentiate Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

Harder integration Integrating Composite Functions we get
You have 1 minute to come up with the rule. Integrating Composite Functions Harder integration we get

Integrating Composite Functions
1. Add one to the power. 2. Divide by new power. 3. Compensate for bracket. Integrating Composite Functions Example :

Integrating Composite Functions
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

Integrating Composite Functions
Example

Integrating Composite Functions
Example

Integrating Functions
1. Add one to the power. 2. Divide by new power. 3. Compensate for bracket. Integrating Functions Example Integrating So we have: Giving:

Calculus Revision Standard Integral (from Chain Rule) Back Next Quit
Integrate Standard Integral (from Chain Rule) Back Next Quit

Calculus Revision Integrate Straight line form Back Next Quit

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

Calculus Revision Integrate Straight line form Back Next Quit

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

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

Calculus Revision Evaluate Back Next Quit

Calculus Revision Find p, given Back Next Quit

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

Given the acceleration a is:
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 Back Next Quit

Integrating Trig Functions
Integration is opposite of differentiation Worked Example

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

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

Integrating Trig Functions
Integrate outside the bracket Keep the bracket the same Compensate for inside the bracket. Integrating Trig Functions Example Integrate Re-arrange

Integrating Trig Functions (Area)
Example The diagram shows the graphs of y = -sin x and y = cos x Find the coordinates of A Hence find the shaded area C A S T 0o 180o 270o 90o

Integrating Trig Functions (Area)

Integrating Trig Functions
Example Remember cos(x + y) =

Integrating Trig Functions

Calculus Revision Find Back Next Quit

Calculus Revision Find Back Next Quit

Calculus Revision Find Back Next Quit

Calculus Revision Integrate Integrate term by term Back Next Quit

Calculus Revision Find Integrate term by term Back Next Quit

Calculus Revision Find Back Next Quit

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

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

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

Are you on Target ! Update you log book
Make sure you complete and correct ALL of the Calculus questions in the past paper booklet.