DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS, LOGARITHMIC FUNCTIOS AND EXPONENTIAL FUNCTIONS

Back to: Further Mathematics SS2

DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

The derivative of y = sin x  dy/dx = cos x

The derivative of y = cos x dy/dx = – sin x

The derivative of y= tan x   dy/dx = sec² x\

The derivative of y = sec x dy/dx = secxtanx

The derivative of y = cosec x dy/dx = cosec x cot x

The derivative  of y = cot x dy/dx = – cosec² x

Examples

(1) If  y = cos 2x

dy/dx = – sin2x  x  d/dx ( 2x)

dy/dx = -2 sin 2x

( 2) If y = cos² x

Let u = cos x and y = u²

dy/du= 2u and du/dx= -sinx

dy/dx = 2u x – sinx

dy/dx = – 2 cos x sin x

( 3) If  y = sec 6x

Let u = 6x and y= sec x

du/dx = 6 and dy/du = sec u tan u

dy/dx = 6 sec u tan u

dy/dx  = 6 sec 6x tan 6x

 

EVALUATION

Differentiate the followings : (i)  y = tan 8x   (ii)  y= cot 5x  (iii)  y = sin⁴ x

THE DERIVATIVE OF LOGARITHMIC FUNCTIONS

If   y = log_e x   dy/dx = 1/x(note that log_ex=lnx)

Examples

(1) If  y =log_e ( 3x + 2 )

dy/dx  =dy/du  x  du/dx

Let u  = 3x +2 and y = log_e u

du/dx = 3  and  dy/du  = 1/u

dy/du   = 1/u  x 3  = 3/3x + 2

(2)  If    y = log_e( 4x – 1) ²

Let  u = ( 4x – 1 )² and  y = log_e u

du/dx = du/dv  x  dv/dx where  v = 4x – 1

du/dx =2v  x  4 =  8v

dy/du  =  1/u

dy/dx  =dy/dx  x  du/dx  = 1/u  x 8v

dy/dx  =  8 (4x – 1)/(4x – 1)²

dy/dx  = 8 /4x-1

 

EVALUATION

Differentiate the followings: ( i)   y = log_e 8x    (ii)   y = ln ( 6x + 9 )³   (iii)   y = ln (3x² – 5x +6)

THE DERIVATIVE OF EXPONENTIAL FUNCTIONS

If  y  = e^xdy/dx = e^x

Examples

(1) If   y = e^2x

dy/dx = dy/du  x  du/dx

u =  2x  and   y = e^u

du/dx = 2  and  dy/du = e^u

dy/dx = e^u  x  2

dy/dx = e^2x  x  2

dy/dx = 2 e^2x

(2)  If   y = e^sin4x

dy/dx = dy/du   x du/dx

Let  u = sin 4x  and   y = e^u

du/dx = 4 cos 4x  and  dy/du = e^u

dy/dx  =e^u  x  4 cos 4x

dy/dx = e^{sin x}x  4 cos 4x

dy/dx = 4 e^sin4xcos 4x

 

EVALUATION

Differentiate the followings : (i)  y = e^{tan 7x}  (ii)  y = e^6x   (iii)   y = e^-5sin3x

 

GENERAL EVALUATION

(1) Find the derivative of each of the following functions : (i)   sin³ x  (ii) cosec x²

(2) Find the derivative of each of the following functions ; (i)   log ( x²  -5x + 6 )

(3) Differentiate each of the followings  :   (i) e^{cosec x}    (ii)  e^x  –  e^-x

(4) Differentiate  log ( cos x + sin x )

Reading Assignment : New Further M aths Project   2   page   13o – 137

 

WEEKEND ASSIGNMENT

1)  If  y = log_e ( 1/x)    find  dy/dx    a) 1/x   b) -1/x  c)  1/x²  d) -1/x²

2) If  y = 3 e^5x   find dy/dx   a) 3e^5x  b) 15e^3x   c)  15e^5x  d) 5e^5x

3) If  y = sin 4x  find  dy/dx  a)  4 cos 4x  b) -4cos4x  c)  4sin4x  d) 4tan 4x

4) If y = cot 7x  finddy/dx  a) 7sec² x  b) -7cosec² x  c) -7cosec² 7x  d) 7 tan 7x

5) Differentiate   sin x – cos x  a) sinx + cosx  b) cosx – sinx  c) sinx- cosx  d)   -sinx-cosx

 

THEORY

1) Differentiate the followings ; (i) cos³ x   (ii) sin 4x  (iii)  e^{cos 5x}   (iv) cos4x³

2) Differentiate the followings  : (i)  ln sin x  (ii) log ( x² – 2)  (iii)  log  ( 1 + x )⁴

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