Differentiation

Question 1

Chain rule

Answer 1

Y= (f(x))^n

Dy/dx = n f’(x) (f(x))^n-1

Question 2

Ditty for the product rule

Answer 2

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

Question 3

Ditty for the quotient rule

Answer 3

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

Question 4

E^x

Answer 4

E^x

Question 5

E^f(x)

Answer 5

F’(x) e ^f(x)

Question 6

Lnx

Answer 6

1/x

Question 7

Ln f(x)

Answer 7

F’(x)/ f(x)

Question 8

A^x

Answer 8

A^x lna

Question 9

A^f(x)

Answer 9

F’(x) Lna af(x)

Question 10

Sin x

Answer 10

Cos x

Question 11

Cos x

Answer 11

-sin x

Question 12

Sin f(x)

Answer 12

F’(x) cos f(x)

Question 13

Cos f(x)

Answer 13

-f’(x) sin f(x)

Question 14

Tan x

Answer 14

Sec^2 x

Question 15

Tan f(x)

Answer 15

F’(x) sec^2 f(x)

Question 16

Cot x

Answer 16

-cosec^2 x

Question 17

Cot f(x)

Answer 17

-f’(x) cosec ^2 f(x)

Question 18

Sec x

Answer 18

Sec x tan x

Question 19

Sec f(x)

Answer 19

F’(x) sec f(x) tan f(x)

Question 20

Cosec x

Answer 20

-cosec x cot x

Question 21

Cosec f(x)

Answer 21

-f’(x) cosec f(x)cot f(x)

Question 22

When do you use the chain rule

Answer 22

When the function you are differentiating is to a power

Question 23

When do you use the product rule

Answer 23

When 2 functions are multiplying by each other

Question 24

When do you use the quotient rule

Answer 24

If one function is dividing another function

Question 25

Small angle approximation for sin theta (radians)

Answer 25

Theta

Question 26

Small angle approximation for tan theta (radians)

Answer 26

Theta

Question 27

Small angle approximation for cos theta (radians)

Answer 27

1- theta^2/2

Question 28

Differentiating sin x using first principles

Answer 28

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

Question 29

Differentiating cos x using first principles

Answer 29

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

Question 30

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

Answer 30

A negative

Question 31

How do you differentiate trig with powers

Answer 31

Use the chain rule

Question 32

Describe how to differentiate y=7^x

Answer 32

``` 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 ```

Question 33

How do you prove how to differentiate tan x

Answer 33

Tan x = sinx / cosx | Then use the quotient rule

Question 34

How do you prove how to differentiate cot x

Answer 34

Cotx = cosx/sinx | Then use the quotient rule

Question 35

How do you prove how to differentiate secx

Answer 35

Y=1/cos x | Then use the chain rule to the power of -1

Question 36

How do you prove how to differentiate cosec x

Answer 36

Y= 1/sin x | Then use the chain rule to the power of -1

Question 37

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

Answer 37

Differentiate with respect to y | Dx/dy

Question 38

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

Answer 38

Find dx/ dy | Then reciprocal

Question 39

Differentiate y= arcsin x

Answer 39

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

Question 40

Differentiate y= arccos x

Answer 40

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

Question 41

Differentiate y= arctan x

Answer 41

1/ 1+x^2

Question 42

What is an implicit equation

Answer 42

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

Question 43

How do you differentiate an implicit equation

Answer 43

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

Question 44

How do you find the turning point of an implicit equation

Answer 44

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

Question 45

When d2y/dx^2 > 0

Answer 45

Minimum turning point

Question 46

When d2y/dx^2 <0

Answer 46

Maximum turning point

Question 47

When dy/dx >0

Answer 47

Increasing function

Question 48

When dy/dx <0

Answer 48

Decreasing function

Question 49

Convex

Answer 49

Minimum point | When d2y/dx^2 > 0

Question 50

Concave

Answer 50

Maximum point | When d2y/dx^2 <0

Question 51

Stationary point of inflection

Answer 51

Dy/dx = 0 | And d2y/dx^2 changes sign

Question 52

Non stationary point of inflection

Answer 52

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

Question 53

What is a point of inflection

Answer 53

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