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Maths Pure Year 2 > Differentiation > Flashcards

Differentiation Flashcards

1
Q

Chain rule

A

Y= (f(x))^n

Dy/dx = n f’(x) (f(x))^n-1

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2
Q

Ditty for the product rule

A

1st x differential of the 2nd + 2nd x differential of the 1st

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3
Q

Ditty for the quotient rule

A

(Bottom x differential of the top - top x differential of the bottom) all divided by the bottom^2

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4
Q

E^x

A

E^x

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5
Q

E^f(x)

A

F’(x) e ^f(x)

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6
Q

Lnx

A

1/x

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7
Q

Ln f(x)

A

F’(x)/ f(x)

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8
Q

A^x

A

A^x lna

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9
Q

A^f(x)

A

F’(x) Lna af(x)

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10
Q

Sin x

A

Cos x

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11
Q

Cos x

A

-sin x

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12
Q

Sin f(x)

A

F’(x) cos f(x)

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13
Q

Cos f(x)

A

-f’(x) sin f(x)

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14
Q

Tan x

A

Sec^2 x

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15
Q

Tan f(x)

A

F’(x) sec^2 f(x)

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16
Q

Cot x

A

-cosec^2 x

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17
Q

Cot f(x)

A

-f’(x) cosec ^2 f(x)

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18
Q

Sec x

A

Sec x tan x

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19
Q

Sec f(x)

A

F’(x) sec f(x) tan f(x)

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20
Q

Cosec x

A

-cosec x cot x

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21
Q

Cosec f(x)

A

-f’(x) cosec f(x)cot f(x)

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22
Q

When do you use the chain rule

A

When the function you are differentiating is to a power

23
Q

When do you use the product rule

A

When 2 functions are multiplying by each other

24
Q

When do you use the quotient rule

A

If one function is dividing another function

25
Q

Small angle approximation for sin theta (radians)

A

Theta

26
Q

Small angle approximation for tan theta (radians)

A

Theta

27
Q

Small angle approximation for cos theta (radians)

A

1- theta^2/2

28
Q

Differentiating sin x using first principles

A

Substitute the f(x+h) with sin (x+h)
This is then the sin angle formula so will be sinxcosh + sinhcosx then -sinx on the end all over h
Factorise them so you have your sin x (cos h-1) on one side and the cosx sinh on the other
Cosx x 1 so f’(x)= cos x

29
Q

Differentiating cos x using first principles

A

Set up using formula
Then get the cos angle formula -cosx
Factories so the cosx (cosh - 1) then - sinx sin h
So you get f’(x)= -sinx

30
Q

If the trig function begins with co what will it differentiate to

A

A negative

31
Q

How do you differentiate trig with powers

A

Use the chain rule

32
Q

Describe how to differentiate y=7^x

A 
Write it as e^ln 7^x
Bring the x down and in front so e^xln7
Differentiate so ln7 e^x ^ln7
Put the x back so ln7 e ^ln7^x
E and th ln cancel out so you’re left with ln7 . 7^x
33
Q

How do you prove how to differentiate tan x

A

Tan x = sinx / cosx

Then use the quotient rule

34
Q

How do you prove how to differentiate cot x

A

Cotx = cosx/sinx

Then use the quotient rule

35
Q

How do you prove how to differentiate secx

A

Y=1/cos x

Then use the chain rule to the power of -1

36
Q

How do you prove how to differentiate cosec x

A

Y= 1/sin x

Then use the chain rule to the power of -1

37
Q

What do you do if the function is given in terms of y

A

Differentiate with respect to y

Dx/dy

38
Q

What do you do if the function is given in terms of y but is asking for dy/do

A

Find dx/ dy

Then reciprocal

39
Q

Differentiate y= arcsin x

A

1/ square root of (1-x^2)

40
Q

Differentiate y= arccos x

A

-1/ square root of (1-x^2)

41
Q

Differentiate y= arctan x

A

1/ 1+x^2

42
Q

What is an implicit equation

A

Where x and y are muddled together and can’t be separated

43
Q

How do you differentiate an implicit equation

A

Differentiate the y with respect to x and put dy/dx on the end
Often uses the product rule as the x and y are stuck together

44
Q

How do you find the turning point of an implicit equation

A

Find dy/dx
Make equal to zero
Always keep the dy/dx on the left, move anything that has this asa factor onto the left, factorise then make it the subject

45
Q

When d2y/dx^2 > 0

A

Minimum turning point

46
Q

When d2y/dx^2 <0

A

Maximum turning point

47
Q

When dy/dx >0

A

Increasing function

48
Q

When dy/dx <0

A

Decreasing function

49
Q

Convex

A

Minimum point

When d2y/dx^2 > 0

50
Q

Concave

A

Maximum point

When d2y/dx^2 <0

51
Q

Stationary point of inflection

A

Dy/dx = 0

And d2y/dx^2 changes sign

52
Q

Non stationary point of inflection

A

Convex section where d2y/dx^2 > 0
Point of inflection where d2y/dx^2 changes sign
Concave section where d2y/dx^2 < 0

53
Q

What is a point of inflection

A

Where f’’(x) changes sign

D2y/dx^2 will equal zero but you need to show that it changes sign before and after this point

Maths Pure Year 2 (12 decks)

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