BC 1/2
Derivatives: The Chain Rule
Name:
This activity sheet is designed to help you to discover a general rule for differentiating the composition
of two or more functions. To begin, fill in each blank with what you believe will be the derivative of the
given function. Then in the space provided, write the derivative provided by the TI-89 or Wolfram
Alpha. Do these problems in the indicated blocks. This means to check your answers against the
calculators for all the #1s first, then move on the #2s, and so on.
On the TI-89, the d( differentiate command, which can be found under the F3 (Calculus) menu,
is used to determine the derivative of a function. .The syntax is d(the function you wish to differentiate,
the variable you are differentiating with respect to). For example, to differentiate the function
y   3x  4  with respect to x, you would enter d((3x+4)^10,x) . Initially, your results and those
of TI-89's may disagree, but hopefully they will begin matching as you do more problems.
10
Your derivative
(1.1)
y   3x  4  
(1.2)
10
TI-89/Wolfram Alpha derivative

y =

y =
y  x 2  11x  1 

y =

y =
(1.3)
y  x3 1

y =

y =
(2.1)
y

y =

y =
(2.2)
y

y =

y =
(2.3)
y

y =

y =


5
2
5x  7  3
5

x 7  11x 5  x
3

x  5x

8

2
Derivatives: The Chain Rule.1
S16
Your derivative
(3.1)
y  3e 5 x 2
(3.2)
y  e 4x
(3.3)
TI-89/Wolfram Alpha derivative

y =

y =

y =

y =
y ex

y =

y =
(4.1)
y  2 7 x

y =

y =
(4.2)
y  53x

y =

y =
(4.3)
y 9
x

y =

y =
(5.1)
y
10

y =

y =
(5.2)
y  e 3x  2
2

y =

y =
(5.3)
ye
3x 2  2

y =

y =
6 2 x
3
4 x2
e 
x 4
4
4
Complete the following statement regarding how you take the derivative of the composite of two
functions.
Chain Rule: If k(x) = f(g(x)), then k(x) =
Derivatives: The Chain Rule.2
S16