3.8 Derivatives of Inverse Trig Functions

Lewis and Clark Caverns, Montana

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y

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y

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x

y( ) 2 0f x x x= ≥

We can find the inverse function as follows:

2y x= Switch x and y.2x y=

x y=

y x=

2y x=

y x=

2df

xdx

=

At x = 2:

( ) 22 2 4f = =

( )2 2 2 4df

dx= ⋅ =

4m =( )2,4

( )1f x x− =

( )1

1 2f x x− =112

1

2

dfx

dx

− −=

1 1

2

df

dx x

−

=

→

To find the derivative of the inverse function:

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y

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4

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x

y

x

y( ) 2 0f x x x= ≥ 2y x=

y x=

2df

xdx

=

At x = 2:

( ) 22 2 4f = =

( )2 2 2 4df

dx= ⋅ =

4m =( )2,4

( )1f x x− =

1 1

2

df

dx x

−

=( )

1 1 1 14

2 2 42 4

df

dx

−

= = =⋅

At x = 4:

( )1 4 4 2f − = =

( )4,21

4m =

Slopes are reciprocals.

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6

4

2

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x

y

x

y

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4

2

0

x

y

x

y 2y x=

y x=

4m =( )2,4

( )4,21

4m =

Slopes are reciprocals.

Because x and y are reversed to find the reciprocal function, the following pattern always holds:

Derivative Formula for Inverses:

df

dx dfdx

x f a

x a

−

=

=

=1 1

( )

evaluated at ( )f a

is equal to the reciprocal of

the derivative of ( )f x

evaluated at .a

The derivative of 1( )f x−

→

A typical problem using this formula might look like this:

Given: ( )3 5f = ( )3 6df

dx=

Find: ( )1

5df

dx

−

Derivative Formula for Inverses:

df

dx dfdx

x f a

x a

−

=

=

=1 1

( )

( )1 1

56

df

dx

−

=

→

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0.5 1 1.5

siny x=

1siny x−=We can use implicit differentiation to find:

1sind

xdx

−

1siny x−=

sin y x=

sind d

y xdx dx

=

cos 1dy

ydx

=

1

cos

dy

dx y=

We can use implicit differentiation to find:

1sind

xdx

−

1siny x−=

sin y x=

sind d

y xdx dx

=

cos 1dy

ydx

=

1

cos

dy

dx y=

2 2sin cos 1y y+ =2 2cos 1 siny y= −

2cos 1 siny y=± −

But2 2

yπ π

− < <

so is positive.cos y

2cos 1 siny y∴ = −

2

1

1 sin

dy

dx y=

−

2

1

1

dy

dx x=

−→

We could use the same technique to find and

.

1tand

xdx

−

1secd

xdx

1

2

1sin

1

d duu

dx dxu

− =−

12

1tan

1

d duu

dx u dx− =

+

1

2

1sec

1

d duu

dx dxu u

− =−

1

2

1cos

1

d duu

dx dxu

− =−−

12

1cot

1

d duu

dx u dx− =−

+

1

2

1csc

1

d duu

dx dxu u

− =−−

1 1cos sin2

x xπ− −= −1 1cot tan2

x xπ− −= −1 1csc sec2

x xπ− −= −

→

Your calculator contains all six inverse trig functions.However it is occasionally still useful to know the following:

1 1 1sec cosx

x− − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

1 1cot tan2

x xπ− −= −

1 1 1csc sinx

x− − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

π