Differentiation

Question 1

How do you differentiate?

Answer 1

Multiply by the power, then decrease the power by one

Question 2

How do you prepare for differentiation?

Answer 2

Change any roots into powers
x must not be on the denominator of any fraction
Any bracket pairs must be expanded

Question 3

What notation or phrases can we also use to represent differentiation?

Answer 3

f’(x)
dy over dx
Rate of change, gradient function, derived function

Question 4

How do we find the gradient of the tangent to a curve at a given value of x?

Answer 4

Differentiate, then substitute x in to find the gradient

Question 5

How do we find the equation of the tangent to a curve?

Answer 5

Differentiate, then substitute x in to find the gradient
Find y-coordinate by substituting x into the original expression
Use y - b = m(x -a)

Question 6

How do we sketch the graph of the derivative?

Answer 6

As dy over dx = 0 at stationary points, the stationary points become the roots in the graph of the derivative

Use the gradient before, between and after stationary points to decide if the graph will be above or below the x-axis

Question 7

A function is increasing when…?

Answer 7

f’(x) > 0

Question 8

A function is decreasing when…?

Answer 8

f’(x) < 0

Question 9

A function is stationary when…?

Answer 9

f’(x) = 0

Question 10

How do you find the stationary points of a function?

Answer 10

At the stationary points f’(x) = 0

Differentiate then solve to find the X values

Substitute into equation to find y values

Use a nature table to determine the nature (max turning point, min turning point)

Question 11

How do you find when a function is increasing or decreasing?

Answer 11

Differentiate then use a nature table

Question 12

How do you show that a function is always increasing/decreasing?

Answer 12

Differentiate then complete the square to show that f’(x) > 0 / f’(x) < 0 for all values of X

Question 13

How do you find the solution to an optimisation question?

Answer 13

Investigate stationary points (and end points if necessary)

Question 14

What do you get if you differentiate distance?

Answer 14

Speed

Question 15

What do you get if you differentiate speed?

Answer 15

Acceleration

Question 16

What do you get if you differentiate sin x?

Answer 16

Cos x

Question 17

What do you get if you differentiate cos x?

Answer 17

-sin x

Question 18

What is the chain rule for differentiation?

Answer 18

f’(g(x)) × g’(x)

Question 19

How do you differentiate the chain rule?

Answer 19

Differentiate around the brackets, then multiply by the derivative of what is inside the brackets

Question 20

What do you get if you differentiate:
sin(ax +b) ?
cos(ax + b) ?

Answer 20

acos(ax + b)
-asin(ax + b)