Differentiation Of Trigonometric Functions, Logarithmic Functions And Exponential Functions

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

• 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 = loge x dy/dx = 1/x(note that logex=lnx)

Examples

• If y =loge ( 3x + 2 ) dy/dx =dy/du x du/dx

Let u = 3x +2 and y = loge u du/dx = 3 and dy/du = 1/u dy/du = 1/u x 3 = 3/3x + 2

• If   y = loge( 4x – 1) ²

Let u = ( 4x – 1 )² and y = loge 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 = loge 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

• 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

• 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

• Find the derivative of each of the following functions : (i) sin³ x (ii) cosec x²
• Find the derivative of each of the following functions ; (i) log ( x² -5x + 6 )
• Differentiate each of the followings : (i) e^{cosec x} (ii) e^x – e^-x
• Differentiate log ( cos x + sin x )

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

WEEKEND ASSIGNMENT

1. If  y = loge ( 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 following ; (i) cos³ x (ii) sin 4x (iii) e^{cos 5x} (iv) cos4x³
2. Differentiate the following : (i) ln sin x (ii) log ( x² – 2) (iii) log ( 1 + x )⁴