Exercise(^) 5H
P1^
5
EXERCISE 5H 1 Use the chain rule to differentiate the following functions.
(i) y = (x + 2)^3 (ii) y = (2x + 3)^4 (iii) y = (x^2 − 5)^3
(iv) y = (x^3 + 4)^5 (v) y = (3x + 2)−^1 (vi) y
x
=^1
(–^233 )
(vii) y = (x^2 − 1)(^32)
(viii) y = (^) (^1 x + x)
3
(ix) y = (^) ()x− 1
4
2 Given that y = (3x − 5)^3
(i) find
d
d
y
x^
(ii) find the equation of the tangent to the curve at (2, 1)
(iii) show that the equation of the normal to the curve at (1, −8) can be written
in the form
36 y + x + 287 = 0.
3 Given that y = (2x − 1)^4
(i) find ddyx
(ii) find the co-ordinates of any stationary points and determine their nature
(iii) sketch the curve.
4 Given that y = (x^2 − x − 2)^4
(i) find ddyx
(ii) find the co-ordinates of any stationary points and determine their nature
(iii) sketch the curve.
5 The length of a side of a square is increasing at a rate of 0.2 cm s−^1.
At what rate is the area increasing when the length of the side is 10 cm?
6 The force F newtons between two magnetic poles is given by the formula
F
r
=^1
5002
, where r m is their distance apart.
Find the rate of change of the force when the poles are 0.2 m apart and the
distance between them is increasing at a rate of 0.03 m s−^1.
7 The radius of a circular fungus is increasing at a uniform rate of 5 cm per day.
At what rate is the area increasing when the radius is 1 m?