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Worked examples and exercises are in the text

PROGRAMME 7

DIFFERENTIATION

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Standard derivatives

Programme 7: Differentiation

As an aide memoir this slide and the following slide contain the list of standard derivatives which should be familiar.

1

( ) /

.ln

1ln

1log

.ln

n n

x x

kx kx

x x

a

y f x dy dx

x nx

e e

e ke

a a a

xx

xx a

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Worked examples and exercises are in the text

Standard derivatives

Programme 7: Differentiation

2

2

( ) /

sin cos

cos sin

tan sec

cot cosec

sec sec .tan

cosec cosec .cot

sinh cosh

cosh sinh

y f x dy dx

x x

x x

x x

x x

x x x

x x x

x x

x x

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Functions of a function

Programme 7: Differentiation

If:

then

For example:

( ) and ( )

. (called the 'chain rule')

y f u u F x

dy dy du

dx du dx

cos(5 4) so cos and 5 4

.

( sin ).5

5sin(5 4)

y x y u u x

dy dy du

dx du dxu

x

STROUD

Worked examples and exercises are in the text

Functions of a function

Programme 7: Differentiation

If:

then

For example:

ln and ( )

1. .

y u u F x

dy dy du dF

dx du dx F dx

ln sin then

1.cos

sincot

y x

dyx

dx xx

STROUD

Worked examples and exercises are in the text

Functions of a function

Products

Quotients

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Functions of a function

Products

Programme 7: Differentiation

If:

then

For example:

y uv

dy dv duu v

dx dx dx

3

3 2

2

.sin3 then

.3cos3 3 sin3

3 cos3 sin3

y x x

dyx x x x

dx

x x x x

STROUD

Worked examples and exercises are in the text

Functions of a function

Quotients

Programme 7: Differentiation

If:

then

For example:

2

u

yv

du dvv udy dx dx

dx v

2

sin3 then

1

( 1)3cos3 sin3 .1

( 1)

xy

x

dy x x x

dx x

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Logarithmic differentiation

Programme 7: Differentiation

Taking logs before differentiating may simplify the working. For example:

Therefore:

2

2

2

sin then ln ln ln(sin ) ln(cos2 )

cos2

ln cos 2sin 2 1cot 2 tan 2 .

2 sin cos2 2

x xy y x x x

x

d y x x x x dyx x

dx x x x y dx

2 sincot 2 tan 2

cos2 2

dy x x xx x

dx x

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Implicit functions

Programme 7: Differentiation

There are times when the relationship between x and y is more involved and y cannot be simply found in terms of x. For example:

In this case a relationship of the form y = f (x) is implied in the given equation.

sin 2xy y

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Worked examples and exercises are in the text

Implicit functions

Programme 7: Differentiation

To differentiate an implicit function remember that y is a function of x. For example, given:

Differentiating throughout gives:

so.

2 2 25x y

2 2 0dy

x ydx

dy x

dx y

STROUD

Worked examples and exercises are in the text

Standard derivatives

Functions of a function

Logarithmic differentiation

Implicit functions

Parametric equations

Programme 7: Differentiation

STROUD

Worked examples and exercises are in the text

Parametric equations

Programme 7: Differentiation

Sometimes the relationship between x and y is given via a third variable called a parameter. For example, from the two parametric equations

the derivative of y with respect to x is found as follows:

sin and cos2x t y t

1 1. 2sin 2 . since

cos /

14sin cos .

cos4sin

dy dy dt dtt

dx dt dx t dx dx dt

t tt

t

STROUD

Worked examples and exercises are in the text

Parametric equations

Programme 7: Differentiation

The second derivative is found as follows:

2

2. . 4sin

. 4sin .

14cos .

cos4

d y d dy dt

dx dx dx dxd dt

tdt dx

tt

STROUD

Worked examples and exercises are in the text

Learning outcomes

Differentiate by using a list of standard derivatives

Apply the chain rule

Apply the product and quotient rules

Perform logarithmic differentiation

Differentiate implicit functions

Differentiate parametric equations

Programme 7: Differentiation