3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana Greg Kelly, Hanford High...

3.8 Derivatives of Inverse Trig Functions

Lewis and Clark Caverns, Montana

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

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:

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.

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

siny x

1siny xWe 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

y

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 x 1 1cot tan2

x x 1 1csc sec2

x x

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 x

1 1 1csc sinx

x