Tutorial - Chapter 2 - Differentiation PDF

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Tutorial 2: Differentiation Topic 2.2.1 : Derivative using first principle 1. By using differentiation from first principle, find the derivatives of the following functions. x a) y  4 x 2 c) y  x  x 2 e) y  x 1 b)

d) y  2 x  1

y  ( x 1)2  1

f) y  x 

 (a) 8x (b) 2(x  1) (c) 1+2x (d) 

1 2x  1

(e) 

1 x2

1

 x  12

(f) 1

2   x 3 

Topic 2.2.2 : Derivative using table 2. Find the derivative of the following functions a) y  3 100 b) y  3 1 e) y  d) y  x 56 x

c) y   f) y  8 4 x 3

 1 (a) 0 (b) 0 (c) 0 (d) 56x 55 (e)  3  2 x2 

(f)

6   1  4 x 

3. Differentiate f ( x)  2 x4  3cos x  sin a with respect to x ( a constant) 8 x 3  3sin x 

4. Find y of the following functions a) y 

1 5 x 3

d) y  a3  2 tan a 2 3x  2 x  1 g) y  x

2 b) y   sin x  e x 3 e) y 

y h)

1 3 3 2 x  x  ln x 2 4 2sin x  3cos x cos x [ (a)

5 4 x 3

1 3 c) y  t 3  2 t 2

f) y  ( x  1)( x 2  2 x) x  4 2x e 1 y  i) ex  1 2 (b)  cos x  e x 3

3 6 (c) t 2  3 2 t

 x2  6 x 6 3 2 3 1 x x   (d) 3a  2sec a (f) (e) x 2 2 x4 9 1 1 x  (i) ex ] (g) (h) 2sec2 x 2 x 2x x 2

5. Differentiate 1 a) y   x7  5  4

b) y  (1 cos x) tan x

1  2 d) y  x  1  x  5   x 



g) y 



e) y 

c) y  (2  x 3)( x 3  x  1)

3 1 7   x6 x5 x2

f) y  (3 x 2  1)2

 3x  2  5  1 h) y    x  x 

8 3  4 x x3 [ (a)

2



7 6 x 4

(b) sec2 x  cos x



i) y  10 x

(c) 6x5  4x3  9x2  2 (d) 3x 2  10x 

 (f) 12 x(3 x 2  1)

(g)

2 x

5 4



9 2x

1

(h) 

5 2

15 12 2   x 6 x7 x2

2

 8x

1 x2

3

(e)

18 5 14   x7 x6 x3

3 2

(i) 5 x

5 2

 12 x

]

Topic 2.3.1 : Chain Rule 6. Find y of the following function by using chain rule. a) y   x  2sin x 

5

d) y  tan  ex  1 g) y 

2 x3

b) y  ln  x2  1

c) y  ecos x 3

e) y  x(1  x )

f) y  3 2 x  4cos  x

h) y  ex

3

2x



5 x 1

(g) 



i) y   3ln  x2  sin x 

2x x2 1 1  1 2 x  (d) ex sec 2  ex  1 (e)   2  x  x2  5 1 x3 2 x 2  (h) (3x  2)e 3 2 x 3 x 2  x  12 [ (a) 5( x  2sin x)4 (1 2cos x)



(c) ecos x 3 sin x

(b)

(f) (i) 

2

3  4sin x 2x

3  2 x  cos x  x2  sin x

7. Use chain rule to compute the derivative and write your answer in terms of x

]

b) y 

a) y  u2  1; u  3x  2

2 ; u  x2  9 2 u

c) y  u 3  3u 2 1; u  x 2  2

[ (a) 6(3x-2)

(b)

8x (c) 6 x5  12 x3 ] 3  ( x 9) 2

8. Differentiate y  (3x  2) 2 (a) By expansion (b) By the product rule (c) By the chain rule [6(3x+2)]

9. Find the derivatives of each function. b) y 

a) y  3 5x6  12 3 4 2(5x  1)

2

c) y  1 

4x  1 2

1 4x 3

5 e) y  cos  2  5 x 2  3

f) y  sin 3 (4 x  1)

g) y  cos x

h) y  tan2 2 x2

i) y  ln( x 3  x)

j) y  3 ln 2x

k) y  ln  (x  1)(2 x  1)

l) y  6ln 2 x  3

d) y 

m) y 

3

2 e2 x

1

n) y 

3 e

10 x 5

[ (a) 3

(e)

5x

6

 12

2

(b)

8x

4x

2

50 x sin  2   5 x2  3 3x  1 x3  x

o) y  e 2 x  e 3x

x

 1

(c) 3

3  1  8x 4  1 3  4x  

(f) 12(sin2 (4 x 1))(cos(4 x 1))

(i) (l)

6 2 x 3

(k)

(j) (m) 

2 e

x

1

(n)

6 e

Topic 2.3.2 : Product Rule 10.

(g)

30x 3 (5 x 4 1) 2  sin

x

4 x cos x

2

(h) 8 x tan(2 x2 ) sec2 (2 x2 )

(d)

Find the derivatives of each function defined as follow

x

4x 1 ( x  1)(2 x  1)

(o) 2 e2x  3e3x ]

a) y   4 x 2  2 (3 x 3  1)

b) y  4 x2 ( x2  1)

d) y  5 x sin x

e) y  x sin(3  x 2)

f) y  sin x tan x

g) y  ln( x sin x)

h) y  x 2e x

i) y  xesin x

k) y  x csc2 x

l) y  sin x tan 3x

x ln x j) y  e



(d) 5 x cos x  5sin x (g)



(b) 2x x2  1

[ (a) 60x4  18x2  8x

(e)

1  cot x x

1

4

c) y   2x  1 (x  1)5 3

5 4

2

(c) 16 x 1 (2 x 1)2 ( x  1)4

(9x2  4)

sin(3  x 2)  x cos(3  x 2)

(f) tan x(sec x  cos x)

sin(3  x2)

(h) x (2  x )e x

(k) csc2 x(1  2 x cot 2 x)

3

1 x  (j) ln x   e x  2 2 (l) sin x tan 3x(9sin x sec 3x  2 tan 3x cos x) ] (i) ( x cos x 1) esin x

Topic 2.3.3 : Quotient Rule

11.

Find the derivatives of each function defined as follow

2 x 5 3x 2

a) y 

x x 1

b) y 

1 x  2x  5

c) y 

d) y 

x cos x

e) y 

sin x 2  sin 2x

1  f) y  cot  2 1  x 

 1 2 x  g) y  ln    3x 

h) y 

e 2 ( x  1)

[ (a)

(e)

1

 x 1

2

3

3x

(b)

2 3x 2

 x  2x  5 3

(2  sin 2 x )cos x  2sin x cos2 x (2  sin 2 x)2

i) y 

(c) 

2

11

 3 x  2

2

e x 1  x2

(d)

cos x  x sin x cos2 x

x 1 2x  1  csc 2  (g) 2 2 2  x (1  2 x ) (1 x )  1 x  2  (1 x )  x (3 x  1) 3 x e (i) e (h) ] 3 (1  x 2 ) 2 ( x  1) (f)

Topic 2.5 : Parametric Differentiation 12.

Find (a)

dy dx in terms of t for the following parametric equations x  3t 2  3t 3 and y  t (t 2  3)

(c) x  3sin t and y  e t

[(a)

13.

t2  1 t (2  3t )

t and y  t 2  1 t 1 x  5cos3 t and y  7sin3 t

(b) x  (d)

(b) 2t (t  1)2

(c)

et 3cos t

(d) 

7 tan t ] 5

Given x  2t and y  4  4t  4t 2 dy (a) Find by using parametric differentiation dx dy (b) y in term of x and hence find dx [(a) 4t  2 , (b) 2  2x ]

14.

dy for the following functions dx (a) x  t 2  2t y  t 3  3t 3t 3t 2 y (b) x  3 1t 1 t 3

Find the

t 4 t 2

[(a) 15.

15 4 , (b) ] 2 5

A curve has the following parametric equation x t 

1 1 and y  t  t t

t 0

dy Find the coordinates of the point when dx  0

[ (0, 2) (0, 2) ] 16.

Find

dy when dx

x  cos 2 and y  2  sin 2

[  cot 2 ]

Topic 2.5 : Higher Derivatives

17.

Find the first and second derivatives of the following function

(a) f ( x)  x 5  6 x 2  7 x x 3 x  2x (g) y  x  3 x

(d) f ( x) 

y  x2 cos x

(b) y

(e)

2

x 1 x

(c)

y  3x 2  1

x  y 1

(f)

(h)

[ Ans: (a) f ( x)  20 x3  12 (b) y  (2  x2 ) cos x  4 x sin x (c) y  

y  

2x 3  18 x2  36x  24 (x 2  2x ) 3

(e) y 

2 (1  x)3

(f) y 

1  32 x 2

(g) y 

3

3 x

1 4 x3



2

3 2

1 

(d)

2 9 3 x5

(h) ]

3

18.

Find

d y for y  2 x5  3 x3  4 x 1 dx 3

[120 x2 18 ]

Topic 2.6 : Implicit Differentiation 19. (a)

Find the

dy by implicit differentiations dx

1 1  1 x y

(c) xy  25 (g) x 3  y 3  1 (i) x  3 xy  2

3

(b)

3 y2  2 x2  2 xy

(e)

x2  3 xy  y2  15

(h)

xy  y2  1

(c) y  x sin2 y  3xy

(f)

at the point (0,-1)

y

y 1 y 4x 2 y sin 2 y  3 y (c) ,  2 (b) 1 x x 6 y  2x 1 2x sin y cos y  3x 2

[Ans: (a)

( x  y) 2 ( x  y ) 2 1

20.

Find the

( x  y)3  3 y  3

dy by implicit differentiations dx

x2 (g) 2 y

1 (h)  2

(d)  y x

(e)

(i) y  

2 x  3 y 3x  2 y

3(2 x  3 y) 3 y 2 1 9x 3 y 2

(f)

]

(a) xy  2 y  3

(b)

2 x2  xy  2 y  5

(c)

(d) xe y  ye x  2x

(e)

xy  sin(x  y )

(g) exy  ln y 2  x

(h)

2 xy  3 y  x2

(i) x 2  y 2  25

(k)

x 2  y  x3  y 2

(n)

sin( x  y)2  y

(q)

(2 x  y)3  x

(m)

x 2 y  e 2 x y 2  2x

(p)  x  2 y   y 2

1 2

1

x 2y

2

x

 x2  ln    x y 

(f) (i)

3

1 5 y (l) x 3  y 3  xy

x

(o) (r)

2 x  y2  4

xy  x  y  2

1

3(2 x  1) y 3 y y  4x 2  ey  yex cos( x  y)  y (b) (c) (d) (e) (f) 1 y x 2 xe  e x 2 2 x x  cos( x  y ) 4x 3 x2  y x 2( x  y) x (3x  2) y (2  x) y (1  yexy ) 2 (g) (h) (i) (j) (k) (l) y 1 2 y y x 3y2 x 2 x 3 xye xy  2

[Ans: (a)

(m)

2(1  xy  e2 x y2 ) x 2  2 ye 2 x

(n)

2( x  2 y ) 2( x  y )cos(x  y )2 1 (o) (p) (q) 2 1 2( x y) cos( x  y ) 1 4( x  2 y ) 2 y 2x

1 2y x  1  2 (r) ] 2 3(2 x  y ) 2 x (1  x ) 21.

If xy  y 2  1 , use implicit differentiation to find

d2 y at the point (0,1) dx 2

1 [ ] 4...