
Calculus
Chain Rule Practice
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Differentiate each function with respect to x.

2) y  x  
1) y( x  ) 
3) f ( x)( x  )

x
5) y( x)( x  ) 
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4) y  x   ( x)
 x  
6) f ( x)
 x  
(
-1-
)

Worksheet by Kuta Software LLC

(
7) f ( x)

x 
x  
)



8)
9) ysec  x 
x  
 x
10) f ( x)( x  )csc  x 

11) f ( x)cos  x    x  
13) f ( x)
f ( x)
 x  
tan  x 
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12) f ( x)sin  x 
14) ycot
-2-

 x  
Worksheet by Kuta Software LLC

Answers to Chain Rule Practice
1)

dy
( x  ) x 
dx
=  x  ( x  )

dy 
2)
 ( x  )   x 
dx 
x
=

( x )

( x)

 x   x  
3) f ' ( x)( x  )
=
( x)







( x)  x 






dy
4)
( x  )  ( x) ( x  )   x 

dx

( x) ( x   x)
=
( x

)


dy
( x)( x  )   x( x  )  
dx
( x   x)
=
( x  ) 

( x  ) x  ( x  ) x 
 x  
6) f ' ( x)

 x  
( x  ) 
5)
(
=
)
 x  ( x  )  ( x   x  )
( x  ) 



( x  ) x  ( x  ) x
 x 
7) f ' ( x)  

 x 
( x  ) 
x( x   x  )
=
(


)



( x ) ( x )



8) f ' ( x)
=



( x) ( x  )   x( x  ) 

9)
( x) 
dy
sec  x  tan  x   x 
dx
=  x  sec  x  tan  x 
 x   x




( x) ( x )
10) f ' ( x)( x  )csc  x  cot  x   x  csc  x   x 
= x  csc  x  ( x  cot  x   xcot  x  )
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Worksheet by Kuta Software LLC





11) f ' ( x)cos  x  ( x  )   x  ( x  )  sin  x   x


x( x sin  x  sin  x   xcos  x  )
=




( x )
12) f ' ( x)cos  x   x 
=  x  cos  x 
tan  x   x  ( x  )sec   x   x 
(
)
13) f ' x 
tan   x 
x  (tan  x   x  sec   x   xsec   x  )
=
tan   x 




dy
14)
csc  ( x  )   ( x  )   x 

dx

=


 x csc ( x )
( x

)




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Worksheet by Kuta Software LLC