http://www.ck12.org Chapter 2. Derivatives
Finddy/dxfory=sin^3 x.
Solution:
The function can be written asy= [sinx]^3 .Thus
dy
dx=^3 [sinx](^2) [cosx]
=3 sin^2 xcosx
Example 4:
Finddy/dxfory=5 cos( 3 x^2 − 1 ).
Solution:
Letu= 3 x^2 − 1 .By the chain rule,
d
dx[f(u)] =f
′(u)du
dx
wheref(u) =5 cosu.Thus
dy
dx=^5 (−sinu)(^6 x)
=− 5 ( 6 x)sinu
=− 30 xsin( 3 x^2 − 1 )
Example 5:
Finddy/dxfory= [cos(πx^2 )]^3.
Solution:
This example applies the chain rule twice because there are several functions embedded within each other.
Letube the inner function andwbe the innermost function.
y= (u(w))^3
u(x) =cosx
w(x) =πx^2.
Using the chain rule,
d
dx[f(u)] =f
′(u)du
dx
d
dx[u
(^3) ] = d
dx[cos
(^3) (πx (^2) )]
=dxd[cos(πx^2 )]^3
= 3 [cos(πx^2 )]^2 [−sin(πx^2 )]( 2 πx)
=− 6 πx[cos(πx^2 )]^2 sin(πx^2 ).
Notice that we used the General Power Rule and, in the last step, we took the derivative of the argument.