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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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**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

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**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

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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 < 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

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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 Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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**Calculus Revision Back Next Quit Differentiate Straight line form**

Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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**Calculus Revision Back Next Quit Differentiate Straight line form**

Chain Rule Simplify Back Next Quit

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**Calculus Revision Back Next Quit 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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**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:

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**The Chain Rule for Differentiating Trig Functions**

Example

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**The Chain Rule for Differentiating Trig 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 Back Next Quit Differentiate Straight line form**

Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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**Calculus Revision Back Next Quit Differentiate Straight line form**

Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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**Harder integration Integrating Composite Functions we get**

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

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**Integrating Composite Functions**

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

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**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

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**Integrating Composite Functions**

Example

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**Integrating Composite Functions**

Example

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**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:

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

Integrate Standard Integral (from Chain Rule) 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) Back Next**

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) Back Next**

Find Back Next Quit

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

Evaluate Back Next Quit

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Calculus Revision Evaluate Back Next Quit

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Calculus Revision Find p, given Back Next Quit

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**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

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**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

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**Integrating Trig Functions**

Integration is opposite of differentiation Worked Example

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**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

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**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

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**Integrating Trig Functions**

Integrate outside the bracket Keep the bracket the same 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 Find the coordinates of A Hence find the shaded area C A S T 0o 180o 270o 90o

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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 Back Next Quit

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Calculus Revision Find Back Next Quit

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Calculus Revision Find Back Next Quit

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Calculus Revision Integrate Integrate term by term Back Next Quit

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Calculus Revision Find Integrate term by term Back Next Quit

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Calculus Revision Find Back Next Quit

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**passes through the point**

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

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**passes through the point**

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

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**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

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**Are you on Target ! Update you log book**

Make sure you complete and correct ALL of the Calculus questions in the past paper booklet.

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