Techniques of Differentiation


20


3. Differentiate the following with respect to x: (a) y = 5 − x 2


(b) y = 4x + x 2 __


__ 2


(c) y = − x 3 + 2 x 2 ___


3 − x 3


(d) y = 4 (6 − x − x 3 ) (e) y =


__________ 5


(x − x 2 + 2) 4. (a) Find (b) Find (c) Find


___ dx if y = (x − 1)(x + 2). dy


dy


___ dx if y = (2x − 3)(4x + 5). dy


(d) Find ds __


(e) Find


___ dv if p = (3v − 2 ) 2 .


(f) Find dl __


dr if l = r 2 − 1 _____


5. (a) If y = x 2 + 5x, fi nd


(c) If f (x) = x 2 __


r − 1 .


___ dx at x = −2.


dy


(b) If s = 2 t 2 + 7t − 3, fi nd ds __


dt at t = 3. 4 − 2x + 1, fi nd f ʹ(8). (d) If y = x 3 − 4 x 2 + 5x − 2, fi nd dy


___ dx and


(f) Find


___ dx at x = −2 and


___ d x 2 at x = −2


d 2 y


___ d x 2 .


d 2 y


(e) If f (x) = 5 x 3 − 4 x 2 + 7x − 3, fi nd f ʹ(2) and f ʹʹ(−1). dy


if y = 3 x 3 − 5 x 2 + 4x − 2.


6. (a) Find the rate of change of y with respect to x at x = 1


_ 2 if y = 4 x 2 − 6x + 1.


(b) Find the slope and the equation of the tangent to the curve y = x 2 − 5x + 1 at x = 2.


(c) Find the slope and the equation of the tangent to the curve y = f (x) = x 3 − 2 x 2 + 3x − 2 at x = −1.


dt if s = (t − 1)(t + 1) t. dp


5 − x + 1 __


2


7. The air resistance R in Newtons (N) to a body moving with speed v in metres/second is


given by R = 3v 2 ___


70 . Find the rate of change of R with respect to v when v = 7 m/s. 8. (a) For f (x) = 3 x 2 − 5x + 7, fi nd x if f ʹ(x) = 1.


(b) If g (x) = x 3 − 12x + 7, fi nd x for which g ʹ (x) = 0.


(c) If y = x 3 __


3 − x 2 __


2 + x, solve


___ dx if y = x (x − 1)(2x + 1).


T =


_________ 24


___ dx = 3.


dy


9. On a certain day, the temperature T in 0 °C in an arid region was given by 5t (24 − t)


, where t is the time in hours and t = 0 corresponds to midnight, 0 ≤ t < 24.


Find: (a) the temperature at 2 a.m., correct to one decimal place,


(b) the temperature at 3 p.m., correct to one decimal place,


(c) the time at which the temperature was 0 °C,


(d) the rate of change of the temperature with time at 4 p.m., correct to two decimal places,


(e) the time at which the rate of change is 0 °C/h.


10. A spherical balloon is being blown up. Find the rate of change of the volume V (m3) with respect to the radius r (m) when r = 3,


if V = 4 _


3 π r 3 .


11. (a) Find a if the slope of the tangent to y = x 2 − ax + 7 at x = −4 is 12.


(b) Find the slope of the tangent to the curve y = x 3 − 7 x 2 + x − 3 at x = 1.


(c) Find a and c if y = ax + c is a tangent to the curve y = − x 3 + 3x − 2 at x = 2.


305


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