Differentiation

Question 1

given a curve with equation y=f(x), an expression for f’(x) is called…

Answer 1

the derivative

Question 2

how to differentiate

Answer 2

multiply by the power and reduce the power by 1

Question 3

for an expression of the form y=…, we denote the derivative with respect to x by

Answer 3

dy/dx

Question 4

it is important before you differentiate that…

Answer 4

all brackets are multiplied out and there are no fractions with an x term in the denominator

Question 5

how else can you write the square root of x?

Answer 5

x to the power of a half

Question 6

how else to write 1/x squared

Answer 6

x to the power of negative 2

Question 7

in a fractional indice, what are the numerator and demoninator?

Answer 7

numerator - power

denominator - root

Question 8

how to differentiate an expression with several terms

Answer 8

differentiate each term seperately

Question 9

the derivative of an x term is always…

Answer 9

constant

eg: 6x = 6

Question 10

the derivative of a constant (eg: 3, 20, π) is always

Answer 10

zero

Question 11

what can x/5 also be written as?

Answer 11

1/5 x

Question 12

how to differentiate a fraction with multiple terms on the numerator

Answer 12

split the fraction up

Question 13

how to denote answer if you are differentiating with respect to t?

Answer 13

d/dt

Question 14

when differentiating with respect to a certain variable, all other variables are…

Answer 14

treated as constants

Question 15

what is the derivative of px squared with respect to p?

Answer 15

x squared

Question 16

the derivative of a function describes its…

Answer 16

rate of change

Question 17

how to find the rate of change of a function when x=3

Answer 17

differentiate the function and then sub 3 in for x

Question 18

what is the velocity v of an object defined as?

Answer 18

the rate of change of displacement s with respect to time t

d=ds/dt

Question 19

acceleration a is defined as the…

Answer 19

rate of change of velocity with respect to time

a=dv/dt

Question 20

we can determine the gradient of a curve, at a particular point, by…

Answer 20

considering a straight line which touches the curve at a point

the tangent

Question 21

to work out the equation of a tangent we need to know which two things?

Answer 21

a point, of which at least one coordinate is given

the gradient, which is calculated by differentiating and substituting in the value of x at the required point.

Question 22

when finding the equation of the tangent to a curve, dy/dx=….

Answer 22

gradient

Question 23

steps on how to find the equation of a tangent to a curve?

Answer 23

1. differentiate the equation of the curve
2. sub in the value of x given in the question to the gradient expression
3. if necessary, use x value in the question to work out y value by subbing into equation
4. y-b=m(x-a)

Question 24

how to find coordinates of points on curve where the tangent has a certain gradient?

Answer 24

1. differentiate to get gradient of tangent
2. find where it is equal to specified value
3. find y coords using equation and subbing in calculated values of x

Question 25

when is a curve said to be strictly increasing?

Answer 25

when dy/dx > 0 *because tangent will slope upwards from left to right as gradient is positive*

Question 26

when is a curve said to be strictly decreasing?

Answer 26

when dy/dx < 0 *because tangent will slope downwards from right to left as gradient is negative*

Question 27

how to determine whether a curve is increasing or decreasing at a given value of x

Answer 27

1. differentiate 2. sub in x value 3. evaluate if < or > than 0

Question 28

how to prove that a curve is never decreasing

Answer 28

if the expression for the derivative is (something)^2 because the result of squaring a number is always greater than or equal to zero dy/dx > or equal to 0 for all values of x

Question 29

at some points, a curve may be neither increasing nor decreasing - we call these points...

Answer 29

stationary points

Question 30

what are the four possible stationary points?

Answer 30

1. maximum turning point 2. minimum turning point 3. rising point of inflection 4. falling point of inflection

Question 31

how to show where an expression for a straight line is increasing or decreasing

Answer 31

1. differentiate 2. identify y-intercept 3. calculate where it cuts the x-axis (when y=0) 4. draw sketch

Question 32

steps to determine the nature of stationary points

Answer 32

1. differentiate the function 2. state that stationary points occur when f'(x) = 0 and solve for x coords of stationary points 3. find y coords by subbing x values into original equation 4. use nature table to calculate f'(x) for values slightly lower and higher than sps 5. evaluate nature of stationary points based on if the values are positive or negative

Question 33

in order to sketch a curve, what needs to be found first?

Answer 33

1. roots (when y=0) 2. y-intercept (when x=0) 3. stationary points and their nature

Question 34

d/dx (sin x) =

Answer 34

cos x

Question 35

d/dx (cos x) =

Answer 35

-sin x

Question 36

the rules for differentiating sin and cos only work if...

Answer 36

in radians

Question 37

the chain rule

Answer 37

differentiate the outer functions, the bracket stays the same, then multiply by the derivative of the bracket

Question 38

closed intervals: the maximum and minimum y-values can either be...

Answer 38

1. stationary points | 2. end points of the closed interval

Question 39

how to work out points at closed intervals

Answer 39

sub x values given in the question into original equation

Question 40

things to remember when drawing a derived graph

Answer 40

- all stationary points on the original curve become roots - wherever the original curve is strictly decreasing, the derivative is negative and will lie below the x-axis - wherever the original curve is strictly increasing, the derivative is positive and will lie above the x-axis

Question 41

what does the derived graph of a quadratic look like?

Answer 41

a linear (straight line)

Question 42

what does the derived graph of a cubic look like?

Answer 42

a quadratic

Question 43

what does the derived graph of a quartic look like?

Answer 43

a cubic

Question 44

what is the process of finding optimal values known as?

Answer 44

optimisation

Question 45

steps for optimisation questions

Answer 45

1. differentiate 2. =0 for sps 3. find x value(s) for sps 4. nature table 5. if you get two answers, decide which one would not be possible (ie: negative *look at diagram*)