(^10) „„„„„ A Textbook of Engineering Mechanics
Following are the rules to remember the above 30 formulae :
Rule 1. Trigonometrical ratio changes only when the angle is (90° – θ)or (90° + θ). In all
other cases, trigonometrical ratio remains the same. Following is the law of change :
sin changes into cos and cos changes into sin,
tan changes into cot and cot changes into tan,
sec changes into cosec and cosec changes into sec.
Rule 2. Consider the angle θ to be a small angle and write the proper sign as per formulae
8 to 11 above.
1.22. DIFFERENTIAL CALCULUS

1. 
   

   d
   dx
   is the sign of differentiation.

   
   


2. 
   

   ddd()xnxnn–1; () 8 , () 1x x x 8 7
   dx dx dx

   
   

   (i.e., to differentiate any power of x, write the power before x and subtract on from the
   power).

   
   


3. 
   

   () 0; (7) 0

   
   

dd

C

dx dx

==

(i.e., differential coefficient of a constant is zero).

4. (.)..

ddvdu

uv u v

dx dx dx

=+

..,

(1 sec )

(2

i e Differential

coefficient of st function Differential coefficient of ond function

product of any nd function Differential coefficient of first funct

two functions

⎡⎤

⎢⎥×

⎢⎥=

⎢⎥+×

⎢⎥

⎢⎥⎣⎦

ion)

⎡⎤

⎢⎥

⎣⎦

5. 2

.–.

du dv

vu

du dx dx

dx v v

⎛⎞

⎜⎟=

⎝⎠

()

(

i.e., Differential coefficient of Denominator × Differential coefficient of numerator

two functions when one is = – Numerator × Differential coefficient of denominat

divided by the other

⎡⎤

⎢⎥

⎢⎥

⎢⎥⎣⎦

or)

Square of denominator

⎡⎤

⎢⎥

⎢⎥

⎢⎥⎣⎦

6. Differential coefficient of trigonometrical functions

(sin ) cos ; (cos ) – sin

dd

xx x x

dx dx

==

dd(tan )xx xsec (^22) ; (cot ) – cosec x
dx dx

(sec ) sec. tan ; (cosec ) – cosec. cot
dd
xxx x xx
dx dx

(i.e., The differential coefficient, whose trigonometrical function begins with co, is negative).