Көрсеткіштік функцияның туындысы. Алгебра, 11 сынып, презентация.

P2 Chapter 9 :: Differentiation

jfrost@tiffin.kingston.sch.uk

www.drfrostmaths.com

Last modified: 6th September 2018

www.drfrostmaths.com

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1:: Differentiate trigonometric, exponential and log functions.

2:: Use chain, product and quotient rules.

3:: Differentiate parametric equations.

4:: Implicit Differentiation

5:: Rates of change

Proof

Yes, you could be tested on this!

Use addition formula.

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Fro Memory Tip: The way I picture this in my head is this:

where moving ‘down’ is differentiating and moving up is integrating. If it’s the ‘wrong’ function (differentiating cos or integrating sin) the sign changes.

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Fro Note: This is not a rule in itself but a specific case of the ‘chain rule’ which we’ll see later this chapter.

Quickfire Questions:

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Pearson Pure Mathematics Year 2/AS

Page 234

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i.e. Scaling the input of a log does not affect the gradient.

This variant occurs regularly in exams!

i.e. When we differentiate an exponential function, we multiply by ln of the base.

We will prove this after we cover implicit differentiation.

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‘Meatier’ Example:

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bun pun intended

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This one is a bit of a headscratcher (pun intended)

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Pearson Pure Mathematics Year 2/AS

Pages 236-237

Functions can interact in different ways…

How to differentiate

Composite Function

The Chain Rule (Ex9C)

Product of Two Functions

The Product Rule (Ex9D)

Division (i.e. “Quotient”) of Two Functions

The Quotient Rule (Ex9E)

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The chain rule allows us to differentiate a composite function, i.e. a function within a function.

Full Method:

This represents the ‘outer’ function.

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Doing it mentally in one go:

(aka the ‘bla method’)

Now differentiate the inner function, i.e. the ‘bla’.

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Outer Function Differentiated

Inner Function Differentiated

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Outer Function Differentiated

Inner Function Differentiated

This is consistent with what we learned earlier in the chapter.

Fro Tip: When differentiating a trig function to a power, always rewrite using a bracket first.

Here we had to use the Chain Rule twice!

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C3 June 2011 Q1a

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Pearson Pure Mathematics Year 2/AS

Pages 239-240

Extension:

As mentioned previously, the product rule is used, unsurprisingly, when we have a product of two functions.

This is quite easy to remember. Differentiate one of the things but leave the other. Then do the other way round. Then add!

Since addition is commutative, it doesn’t matter which way round we do it.

Sub into product rule.

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It’s typical to factorise when finding a turning point.

The fact that exponential terms can never be 0 comes up frequently in exams.

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Edexcel C3 June 2013 Q5(c)

Examiner’s Report: “Perhaps the most challenging question in the paper”. And this is on a paper described as having “a level of demand higher than any previous C3 paper”.

So it’s DAMN FRIKKIN’ HARD. 

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Edexcel C3 Jan 2012 Q1a

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Pearson Pure Mathematics Year 2/AS

Page 242

Just as we use the ‘product rule’ to differentiate a ‘product’, we use the ‘quotient rule’ to differentiate a ‘quotient’ (i.e. division).

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Layout just like you would with the Product Rule.

Whenever you have “fraction = 0”, you can just multiply through by denominator.

As per advice earlier this chapter, try to factorise whenever you have 0 on one side.

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Edexcel C3 Jan 2012 Q1a

Edexcel C3 June 2012 Q3

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Pearson Pure Mathematics Year 2/AS

Pages 244-245

Extension

Use quotient rule.

Fro Tip: This is in the formula booklet, but I HIGHLY encourage your to memorise it to save you time in the exam.

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Memory Tip: To memorise the two on the right, notice that when we ‘co’ the two on the left, it ‘co’s each term in the derivative, but also negates it.

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Edexcel C3 June 2013(R) Q5b

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Edexcel C3 Jan 2011 Q8b,c

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Pearson Pure Mathematics Year 2/AS

Pages 249-251

Extension:

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C4 June 2012 Q6

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Pearson Pure Mathematics Year 2/AS

Pages 252-254

MIND-BLOWING MOMENT

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C4 Jan 2008 Q5

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C4 June 2014(R) Q3

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Hint for (b): Solve simultaneously with original equation.

Exam Note: This type of question was quite common.

Pearson Pure Mathematics Year 2/AS

Pages 255-256

In my Year 1 slides you may remember that a point of inflection was where the concavity of a curve changes, i.e. concave to convex or vice versa, or informally, ‘swerving one way to swerving the other’.

We say the curve is concave when it is swerving right: we can see the gradient is always decreasing, i.e. the change in the gradient is negative.

At point of inflection, the gradient is not changing, i.e. the gradient of the gradient is 0. Note that points of inflection (as with this example) are not necessarily stationary points.

In this region the gradient, although initially negative, is still also increasing. The curve is swerving left.

Memorisation tip: Mentally visualise a positive cubic (i.e. uphill) and put ‘concave’ and ‘convex is alphabetical order.

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Pearson Pure Mathematics Year 2/AS

Pages 259-261

Q

Firstly, how would we represent…

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Then by Chain Rule:

Click for Fromanimation

First copy top and bottom into the diagonals…

…Then fill in the gaps with whatever variable you didn’t use.

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Edexcel C4 June 2008 Q3

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It’s common to have a second part of the question where you have to connect further rates.

June 2012 Q2

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Pearson Pure Mathematics Year 2/AS

Pages 263-264

Note: I have skipped out the content on ‘setting up differential equations’ as I think it’s better to do this in the Integration chapter. You therefore won’t be able to yet do Q5, Q8, Q9, Q12, Q14.

Extension:

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