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Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig Functions Differentiation The Chain Rule

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Higher Outcome 2 The Chain Rule for Differentiating To differentiate composite functions (such as functions with brackets in them) we can use: Example

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Higher Outcome 2 The Chain Rule for Differentiating Good News ! There is an easier way. 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.

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Higher Outcome 2 The Chain Rule for Differentiating Example 1. Differentiate outside the bracket. 2. Keep the bracket the same. 3. Differentiate inside the bracket. You are expected to do the chain rule all at once

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

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Higher Outcome 2 Example The Chain Rule for Differentiating

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Higher Outcome 2 Example Re-arrange: The slope of the tangent is given by the derivative of the equation. Use the chain rule: Where x = 3: The Chain Rule for Differentiating Functions

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Higher Outcome 2 Is the required equation Remember y - b = m(x – a) The Chain Rule for Differentiating Functions

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Higher Outcome 2 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

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Higher Outcome 2 Using chain rule The Chain Rule for Differentiating Functions

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Higher Outcome 2 For x < 5 we have (+ve)(+ve)(-ve) = (-ve) Therefore x = 5 is a minimum Is x = 5 a minimum in the (complicated) graph? Is this a minimum? For x > 5 we have (+ve)(+ve)(+ve) = (+ve) The Chain Rule for Differentiating Functions For x = 5 we have (+ve)(+ve)(0) = 0 x = 5

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Higher Outcome 2 The cost of production: Expensive components? Aeroplane parts maybe ? The Chain Rule for Differentiating Functions

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Calculus Revision Back Next Quit Differentiate Chain rule Simplify

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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Calculus Revision Back Next Quit Differentiate Chain Rule

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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

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

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Higher Outcome 2 Trig Function Differentiation The Derivatives of sin x & cos x

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Higher Outcome 2 Example Trig Function Differentiation

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Higher Outcome 2 Example Trig Function Differentiation Simplify expression - where possible Restore the original form of expression

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

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Higher Outcome 2 The Chain Rule for Differentiating Trig Functions Example

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Higher Outcome 2 Example The Chain Rule for Differentiating Trig Functions

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Calculus Revision Back Next Quit Differentiate

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Calculus Revision Back Next Quit Differentiate

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Calculus Revision Back Next Quit Differentiate

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Calculus Revision Back Next Quit Differentiate

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

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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

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Calculus Revision Back Next Quit Differentiate

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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Calculus Revision Back Next Quit Differentiate Chain Rule Simplify

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Higher Outcome 2 Integrating Composite Functions Harder integration we get You have 1 minute to come up with the rule.

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

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Higher Outcome 2 Example Integrating Composite Functions 1. Add one to the power. 2. Divide by new power. 3. Compensate for bracket. You are expected to do the integration rule all at once

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Higher Outcome 2 Example Integrating Composite Functions

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Higher Outcome 2 Example Integrating Composite Functions

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

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

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Calculus Revision Back Next Quit Integrate Straight line form

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

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Calculus Revision Back Next Quit Integrate Straight line form

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

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

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Calculus Revision Back Next Quit Evaluate

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Calculus Revision Back Next Quit Find p, given

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

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Calculus Revision Back Next Quit 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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Higher Outcome 2 Integrating Trig Functions Integration is opposite of differentiation Worked Example

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

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

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Higher Outcome 2 Example Re-arrange Integrate Integrating Trig Functions 1.Integrate outside the bracket 2.Keep the bracket the same 3.Compensate for inside the bracket.

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Higher Outcome 2 Integrating Trig Functions (Area) Example A The diagram shows the graphs of y = -sin x and y = cos x a)Find the coordinates of A b)Hence find the shaded area C A S T 0o0o 180 o 270 o 90 o

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Higher Outcome 2 Integrating Trig Functions (Area)

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Higher Outcome 2 Example Integrating Trig Functions Remember cos(x + y) =

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Higher Outcome 2 Integrating Trig Functions

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Calculus Revision Back Next Quit Find

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Calculus Revision Back Next Quit Find

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Calculus Revision Back Next Quit Find

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Calculus Revision Back Next Quit Integrate Integrate term by term

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Calculus Revision Back Next Quit Find Integrate term by term

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Calculus Revision Back Next Quit Find

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

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

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

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Higher Outcome 2 Are you on Target ! Update you log book Make sure you complete and correct ALL of the Calculus questions in theCalculus past paper booklet.

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