© annie patton differentiation of inverse trigonometric function next slide

© Annie Patton

Differentiation of Inverse Trigonometric Function

1sin

1sin 1tan

1tan

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Aim of Lesson

To establish what an inverse trigonometric function is, then differentiate the inverse of sin

and tan from first principles and use these formulas.

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Inverse Trigonometric Function

1, then sin2 2

If Sin

It is the inverse of sin.

1

1

cos is the inverse of cos and

tan is the inverse of tan .

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1for the differentiation of sinx

Special rulea

1sin

sin

xy

ax

ya

sin

cos

x a y

dxa y

dy

2 2

2 2

1 1

cos cos

1

(1 sin )

dy

dx a y a y

a y

a is a constant

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-1

2 2

xdsindy 1a= =

dx dx a -x

This formula is given in the tables.

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1Differentiation of y=sin 4x

1sin 4y x 1

2 2

(sin ) 1with

x=4x and a=1

xd

aComparingdx a x

1

2 2 2 2 2

u=4x

(sin ) 1 1 14

1 ( ) 1 (4 ) 1 16

Let

du d u

dx du u x x

1

2 2

sin 1 44

1 16 1 16

dy d u du

dx du dx x x

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1for the differentiation of tanx

Special rulea

a is a constant

-1 xy=tan

a

2 2

dy 1 1 = =

dx a sec y a (1 + tan y)

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2

xThen tan y=

ax=a tan y

dx=a sec y

dy

2 2 2 2 2

2

1 1

( )(1 )

dy a

a a xdx a xxa

aa

This formula is given in the tables.

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Differentiate of y=tan-14x

u=4x 4du

Letdx

1

2 2

tan

1 1

1 1 16

y u

dy

du u x

2 2

1 4.4

1 16 1 16

dy dy du

dx du dx x x

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1 y=tan , find the value of at x= 2.dy

Given xdx

Leaving Certificate Higher No 6b(i) Paper 1 2004

1

2

tan

1

1

y x

dy

dx x

2

x= 2

1 1 1

1 2 31 ( 2)

At

dy

dx

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1 1 y=sin 10 , find the value of at x= .

20

dyGiven x

dx

Leaving Certificate Higher No 7 b (1) Paper 1 2002

1sin 10

u=10x

10

y x

Let

du

dx

2 2

1 1010

1 100 1 100

dy dy du

dx du dx x x

1 10 10 20at x= ,

20 100 3 31

400 4

dy

dx

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1

2 2

sin

1 1

1 1 100

y u

dy

du u x

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1 1

' '

2f(x)=tan and g(x)=tan , for x>0.

2

f ( ) g ( ).

xLet

x

Find x and x

'2

2( )

4f x

x

1

22

22

22

u xx

dux

dx x

1 2g(x)=tan

x

1

2

2

tan

1 141 1

y u

dy

du ux

2

2 2 2 2

2

1 2 2 2.

4 4 41

dy dy du x

dx du dx x x x xx

'2

2( )

4g x

x

Leaving Certificate Higher No 6c (i) Paper 1 2007

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Homework. Differentiate the following:

6.

-1

-1

-1

-1

-1

-1

1. y= sin 5x

2. y= tan 7x

x3. y= sin

2x

4. y=2x sin2

x5. y= tan

1+xy= tan (cos x)

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Conclusion

1

2 2

sin 1=

xd

adx a x

1 1

for example when y=4x+1. Let u=4x+1 then

sin (4 1) sin= .

And

d x d u du

dx du dx

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Conclusion

1

2 2

tanx

d aadx a x

1 1

for example when y=4x+1. Let u=4x+1 then

tan (4 1) tan= .

And

d x d u du

dx du dx

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•Note the differentiation of cos-1 is not on the syllabus.