Basic Differentiation

The instant rate of change of y with respect to x is written as .

Differentiation

dydx

By long experimentation, it is possible to prove the following:

dydx =

y = x n

nxn–1

If

then • multiply by the power

• reduce the power by

one

How to Differentiate:

Note that describes both the rate of change and the gradient.

dydx

1

The Derived Function

Differentiation

It is also possible to express differentiation using function notation.

=

f (x)

nxn–1

If

then f ′(x)

= x n f ′(x) dy

dxand

mean exactly the same thing written in different ways.

LeibnizNewton

The derived function is the rate of change of the function with respect to .

f ′(x) f (x)x

2

Differentiation of Expressions with Multiple Terms

Differentiation

y =ax m + bx

n + …

• multiply every x-term by the power

How to Differentiate:

amx m + bnx

n + …–1 –1dy

dx =

• reduce the power of every x-term by one

The basic process of differentiation can be applied to

every x-term in an algebraic expession.

ImportantExpressions must be written as the sum of individual terms before differentiating.

3

Example of Basic Differentiation

Differentiation

Find for

dydx

y = 3 x 4 – 5 x

3 + + 9

7x

2

y = 3 x 4 – 5 x

3 + 7 x -2 +

9dydx = 12 x

3 – 15 x 2 – 14

x -3

9 =9 x 0

this disappears because

= 12 x 3 – 15 x

2 –14x

3

4

(multiply by zero)

Differentiating Products

Differentiation

If there are factors to a product, differentiation will followed by using formula.

5

dydx =

dvdx

u +dudx

v

Where, u & v

represents the factors of the product.

Differentiating Quotients

Differentiation

Differentiation of quotient requires following formula.

6

dydx =

dudx

v Where, u & v

represents dividend & divisor respectively.

dvdx

u

v2

Differentiating expressions involve Exponential, ‘e’ and Log

Differentiation 7

dydx of e remains e x x

dydx of e gives ae ax x

●

●

dydx of log x gives● e

1x

dydx of log x gives x log e ● a

1x a

dydx of log x gives x log e ●

10

1x 10

dydx of a gives a x log a ● x

ex

Increasing and Decreasing Curves

Differentiation

The gradient at any point on a curve can be found by differentiating.

8

uphill slopePositive

downhill slopeNegative

Gradient

Ifdydx

> 0 then y is

increasing.

Ifdydx

< 0 then y is

decreasing.dydx < 0

dydx > 0

dydx < 0

If a function is neither increasing or decreasing, the gradient is zero and the function can be described as stationary.

Stationary Points

Differentiation 9

There are two main types of stationary point.

Minimum

Maximum

Turning Points

Points of Inflection

At any stationary point, dydx

= 0

Rising

Falling

Investigating Stationary Points

Differentiation 10

ExampleFind the stationary

point off (x) = x 2 – 8 x + 3

Stationary point

given by

f ′(x) = 2 x – 8

2 x – 8= 0

and determine its

nature.

x= 4

f ′(x) = 0

x 4 – 4+4

slope

The stationary point atis a minimum turning point.

– 0 +f ′(x)

x= 4

Use a nature table to reduce the amount of working.

‘slightly less than

four’

‘slightly more’

gradientis positive

Investigating Stationary Points

Differentiation 11

Example 2

Investigate the stationary

points ofy =4 x 3 – x

4

dydx

=12 x 2 – 4 x

3 = 0

4 x 2 (3 – x ) = 0

4 x 2= 0

x = 0

3 – x= 0

x = 3

or

y = 0 y = 27

x 0 – 0+0

slope

dydx

rising point of inflection

+ 0 +

stationary point at (0,0):

x 3 – 3+3

slope

dydx

maximum turning point

+ 0 –

stationary point at (3, 27):