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Differentiation of Products

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© Annie Patton

Aim of lesson

To learn how to differentiate products.

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© Annie Patton

What is a product?

2 x 3

20 X 46

x(x2 + 3x)

So it is two things multiplied together.

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To get the derivative of a product

• For example (x + 2) (x2 +3x)

• You could multiply it out

• x ( x2 +3x) +2(x2 +3x)

• = x3 + 3x2 + 2x2 + 6x

• =x3 + 5x2 + 6x

( 10 63 2 2x 5x 6x) 3xdy

xdx

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Now for an easier method!!!

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To Differentiate y = (x + 2)(x2 + 3x)

• Let u = x + 2 and v= x2 + 3x

1du

dx

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2 3dv

xdx

2( 2)(2 3) ( 3 )(1)dy

x x x xdx

=3x2+ 10x+6

d(uv) dv du=u +v

dx dx dx

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Product Rule

If y=uv. Then use the formula

dy dv du=u +v

dx dx dx

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'

'

Note if instead of y, you are given f(x),then f (x) is the derivative of f(x).

df(x)f (x)= .

dx

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Why ?

The derivative of y=(x3+3x)(x2+6) equals

(x3+3x)(2x)+(x2+6)(3x2+3)u

dv

dxv

du

dx

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© Annie Patton

2 2 2

2

(2 5)(6 ) (3 6)(2) 12 30 6 12

18 30 12

dyx x x x x x

dxdy

x xdx

Differentiate y=(2x-5)(3x2 +6)

• Let u equal(2x-5) and v equal (3x2 +6).

• Use the formula

• Then

• So

2du

dx

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Start clicking when you want to see the answer.

6dv

xdx

d(uv) dv du=u +v

dx dx dx

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Differentiate y=x3(cos x)

• Let u equal x3 and v equal cos x.

• Use the formula

• Then

• So sin

dvx

dx

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Start clicking when you want to see the answer.

3 2 3 2( sin ) 3 cos sin 3 cosdy

x x x x x x x xdx

23du

xdx

d(uv) dv du=u +v

dx dx dx

Remember from First Principles. Also see tables.

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Proof of Product Rule by First Principles

( ) ( ) ( )f x u x v x

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( ) ( ) ( )f x h u x h v x h

( ) ( ) ( ) ( ) ( ) ( )f x h f x u x h v x h u x v x f(x+h)-f(x)=u(x+h)v(x+h)-u(x+h)v(x)

+u(x+h)v(x)-u(x)v(x)

f(x+h)-f(x) (v(x+h)-v(x)) (u(x+h)-u(x))=u(x+h) +v(x)

h h h

0

f(x+h)-f(x) dv du dflim =u(x) +v(x) =

h dx dx dxh

Leaving Certificate 2000 Higher Level Paper 1 no 6(b)(i)

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Differentiate x (x+2) with respect to x.Leaving Certificate 2006 Higher Level Paper 1 no 6(a)

1

2Let u= x =x Let v=x+2

1-2

du 1 1= x =

dx 2 2 x

dv=1

dx

dy 1= x(1)+(x+2)

dx 2 x

dy x+2= x +

dx 2 x

dy 2x+x+2 3x+2= =

dx 2 x 2 xNext Slide

Start clicking when you want to see the answer.

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2( 2)( 5 ), .dm

If m n n n finddn

Let u=n+2

du=1

dn

Start clicking when you want to see the answer.

2Let v=n +5n

dv=2n+5

dn

2dm=(n+2)(2n+5)+(n +5n)(1)

dn

2 2 22 5 4 10 5 3 14 10dm

n n n n n n ndn

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2

Find theslopeand equation of the

tangent to the curve y=(x +2)(3x) at the point (0,0).

2 2

2

u x

dux

dx

3

3

v x

dv

dx

2

2 2 2

dy=(x +2)(3)+3x(2x)

dxdy

=3x +6+6x =9x +6dx

dyat (0,0) =6

dx

1 1

Equation of the tangent

y-y =m(x-x )

y-0=6(x-0)

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Now differentiate the following exercises by the Product Rule

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2 2

2 3

2

3

2

1. y=x( 6)

2. y=t (4 )

3. y=(x +3x+2)(x +5)

4. y=x cos

5. sin

6. Find theslopeand equation of the

tangent to the curve y=(x -4)(2x+1) at the point (1,-9)

x

t

x

y

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A link to VISUAL CALCULUS

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Click to get more examples for the product rule, but ignore the ones with e in it.

© Annie Patton

Product Rule

d(uv) dv du=u +v

dx dx dx