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8. Differentiate each function. Express your answer in a simplified factored form.
a. f 1x2 1x 42 3 1x 32 6 d. h1x2 x 3 13x 52 2
b. y 1x 2 32 3 1x 3 32 2 e. y x 4 11 4x 2 2 3
c. y 3x2 2x
x 2 1
f. y a x2 4
3
x 2 3 b
9. Find the rate of change of each function at the given value of t. Leave your
answers as rational numbers, or in terms of roots and the number p.
a. b. s(t) = a t p
1
1
2
t 2p
t 6p b 3
s1t2 t 3 14t 52 3 , t 8 ,
10. For what values of x do the curves y 11 x 3 2 2 and y 2x 6 have
the same slope?
11. Find the slope of the tangent to the curve 2 2 at Q2, 1 4 R.
12. Find the equation of the tangent to the curve y 1x 3 72 5 at x 2.
dy
13. Use the chain rule, in Leibniz notation, to find at the given value of x.
dx
a.
b.
c.
d.
y 3u 2 5u 2, u x 2 1, x 2
y 2u 3 3u 2 , u x x 1 2 , x 1
y u1u 2 32 3 , u 1x 32 2 , x 2
y u 3 51u 3 7u2 2 , u Vx, x 4
14. Find h¿122, given h1x2 f 1g1x22, f 1u2 u 2 1, g122 3, and g¿122 1.
A
T
15. A 50 000 L tank can be drained in 30 min. The volume of water remaining in
the tank after t minutes is V1t2 50 000 Q1 t
At what rate,
30 R 2 , 0 t 30.
to the nearest whole number, is the water flowing out of the tank when t 10?
16. The function s1t2 1t 3 t 2 2 1 2 , t 0, represents the displacement s, in
metres, of a particle moving along a straight line after t seconds. Determine
the velocity when t 3.
PART C
17. a. Write an expression for h¿1x2 if h1x2 p1x2q1x2r1x2.
b. If h1x2 x12x 72 4 1x 12 2 , find h¿132.
18. Show that the tangent to the curve y 1x 2 x 22 3 3 at the point 11, 32
is also the tangent to the curve at another point.
19. Differentiate y x2 11 x 3 2
11 x2 3 .
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2.5 THE DERIVATIVES OF COMPOSITE FUNCTIONS
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