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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