Name: _______________________________
Date: ________________________
2.4 The Chain Rule
…BLM 2-9. .
1. Determine the derivative of each function
by using the following methods.
i) simplify first and then differentiate
ii) use the chain rule and then simplify
2 
b) y =  x 4 
3 
4 3
a) y = (3x )
 25x 
c) y =
4
d) y = 32x
2
5
2. Differentiate. Express each answer using
positive exponents.
a) f(x) = (3x – 2)2
b) f(x) = (4x – 1)3
2
2
c) g(x) = (2x + 5) d) g(x) = (4x + 7)–2
e) y = (3x2 – x)3
f) y = (3 – 4x4)–1
3. Express each function as a power with a
rational exponent, and then differentiate.
Express each answer using positive
exponents.
a) f(x) =
2x  5
b) g(x) =
3x 2  4
3x  5x 4
d) p(x) =
5
7x  3x  4
2
4. Express each as a power with a negative
exponent, and then differentiate. Express
each answer using positive exponents.
1
a) f(x) =
2
2x  1
4x
1
 3x

3x 3  x
1
3
3
6. Using Leibniz notation, apply the chain
dy
rule to determine
at the indicated
dx
value of x.
a) y = u2, u = 3x – 5, x = 2
b) y =
c) y =
u , u = x2 – 3x + 1, x = –1
1
, u = 2x3 + 4x, x = 3
u
d) y = u2 –
u , u = –2x2 – x, x = –2
7. Determine the slope of the tangent to the
curve y = (5x2 – 3x)4 at x = 1.

9. Determine the equation of the tangent to
the curve y = (x2 + 3x – 1)3 at x = 0.
10. Determine the point(s) on the curve
y = (4x – 3)2 where the tangent line is
horizontal.
11. Find the second derivative of
y = (x2 + 3x)3.
3
1
c) h(x) =
d) f(x) =

2
 2x  4 

3
b) g(x) =
d) f(x) =
8. Determine the slope of the tangent to the
1
curve y =
at x = –2.
2
3x  2x 3
c) h(x) =

3
3
 
5
2x 4  5x
3
c) f(x) =
12. If f(x) = x3, g(x) = 2x + 1 and h(x) =
determine the derivative of each
composite function.
a) y = f og(x)
b) y = hog(x)
c) y = f og oh(x)
x,
x  5x 2

5. Determine f  3 .
a) f(x) = (3x + 2x – 1)2
b) f(x) = (4x2 + 3)–4
2
13. Use the product rule and chain rule
together to differentiate the following.
a) f(x) = (2x – 3)3(3x – 1)2
b) g(x) = (x2 + 2x)3(4x + 1)–2