CK-12-Calculus

6.5. Derivatives and Integrals Involving Inverse Trigonometric Functions http://www.ck12.org

d

dx

[sin− (^1) u]= √ 1
1 −u^2
du
dx−^1 <u<^1
d
dx
[cos− (^1) u]= √− 1
1 −u^2
du
dx−^1 <u<^1
d
dx
[tan− (^1) u]= 1
1 +u^2
du
dx−∞<x<∞
d
dx
[sec− (^1) u]=∣ 1
∣u∣∣√u^2 − 1
du
dx
∣∣u∣∣> 1
d
dx
[csc− (^1) u]=∣ − 1
∣u∣∣√u (^2) − 1
du
dx
∣∣u∣∣> 1
d
dx
[cot− (^1) u]= − 1
1 +u^2
du
dx−∞<x<∞
Example 3:
Differentiatey=sin−^1 ( 2 x^4 )
Solution:
Letu= 2 x^4 ,so
dy
dx=

√^1

1 −( 2 x^4 )^2 ·(^8 x

(^3) )
=^8 x
3
√ 1 − 4 x 8.
Example 4:
Differentiate tan−^1 (e^3 x).
Solution:
Letu=e^3 x,so
dy
dx=

1

1 +(e^3 x)^2 ·^3 e

3 x

=^3 e

3 x

1 +e^6 x.

Example 5:
Finddy/dxify=cos−^1 (sinx).
Solution:
Letu=sinx.

dy

dx=

√ −^1

1 −sin^2 x

=cos−^1 x.