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METHODS OF DIFFERENTIATION
1. DERIVATIVE OF f(x) FROM THE FIRST PRINCIPLE :
Obtaining the derivative using the definition
,oy ,f(x + ox) - f(x) , dy
LIm - ~ LIm ~ f (x) ~ - is called calculating
Bx-tO ox Sx ~ o ox dx
derivative using first principle or ab initio or delta method,
2, FUNDAMENTAL THEOREMS:
If f and g are derivable function of x, then,
d df dg
(a) -(f±g)~ - ±-
dx dx dx
d df
(b) dx (el) ~ c dx ' where c is any constant
'i
d dg df
~ (e) dx (fg) ~ f dx + g dx known as "PRODUCT RULE"
~
I
1
l
' I,
~
i where 9 *0 known as "QUOTIENT RULE"
j
dy dy du
I
~
(e) Ify ~ ~u)&u~g(x), then dx ~ du' dx knCMll1as "CHAIN RUU"
dy , du
Note In general if y ~ f(u), then dx ~ f (u), dx '
I
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3. DERIVATIVE OF STANDARD RJNCTlONS
,
. , . fIx) "'.
. rex) , " ..
(i) xn nx n- 1
(ii) eX eX
(iii) aX aXtna, a > 0
(iv) Cnx I/x
(v) log.x (I/x) log.e, a > 0 , a " 1
(vi) sinx casx
(vii) casx - sinx
(viii) tanx sec 2x
(ix) secx secx tanx
(x) cosecx - cosecx , colx
(xi) colx - cosec 2x
(xii)
(xiii)
constant
sin- 1 x
° 1
~, - 1 < x < 1
1 - x2
-1
(xiv) C05- 1 x ~. - 1 < x < 1
tan-! x
l _ x2
- -z'
1
,l
(xv)
l +x
xER ,
1
!
sec- 1 x ~, l xl>1 l
(xvi)
I x I xZ_l ,
I
(xvii)
(xviii)
cosec- 1
coC! x
x
-1
I x I ~xz - 1 '
-1
--z,xE R
I x I> 1
I
1
1+x i
j
LOGARITHMIC DIFFERENTIATION:
I!
4.
To find the derivative of :
(a) A function which is the product or quotient of a number of function or
(b) A function of the form [fIx)] 9 (xl where f & 9 are both derivable,
it is convenient to take the logarithm of the function first & then •i•
differentiate,
E
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Mathematics Handbook
5. DIFFERENTIATION OF IMPUCIT FUNCTION:
(a) Let function is 4>(x, y) = 0 then to find dy /dx, In the case of
implicit functions, we differentiate each term w.r.t. x regarding
y as a functions of x & then collect terms in dy / dx together on
one side to finally find dy / dx
dy - 84> / ax 84> 84>
OR dx = 84> / ay where ax & ay are partial differential
coefficient of 4>(x, y) w.r.to x & y respectively.
(b) In answers of dy/dx in the case of implicit functions, both x & y
are present.
6. PARAMETRIC DIFFERENTIATION:
If y = f(El) & x = g(El) where a is a parameter, then dy = dy / dEl
dx dx/d8
7. DERNA11VE OF A FUNC110N W.R.T. ANOTIiER FUNCJ10N :
dy dy / dx f'(x)
Let y= f (x) ; z = 9 (x) , then dz = dz / dx = g '(x)
8. DERNA11VE OF A RJNCJ10N AND ITS INVERSE RJNCJ10N :
If inverse of y = fIx)
x = f-l(y) is denoted by x - g(y) then g(f(x)) = x
g'(f(x»f(x)= 1
9. HIGHER ORDER DERIVATIVE
J
! Let a function y - fIx) be defined on an open interval (a, b). It's
I derivative, if it exists on (a, b) is a certain function fIx) [or (dy / dx)
I! or y' ) & it is called the first derivative of y w. r. t. x. If it happens
that the first derivative has a derivative on (a , b) then this derivative
is called second derivative of y w.r.t. x & is denoted by f"(x) or
t (d 2y /dx 2) or y". Similarly, the 3,d order derivative of y w.r.t x, if it
i
~
~l( d2~)' . It is also denoted by fm (x) or
3
~ exists, is defined by d ;, =
•i" y'" &soon.
dx dx dx
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10. DIFFERENTIATION OF DETERMINANTS :
f(x) g(x) h(x)
If F(x) = I(x) m(x) n(x) , where f, g. h. I , m, n, U, v, ware
u(x) v(x) w(x)
differentiable functions of x, then
t' (x) g'(x) h' (x) f(x) g(x) h(x) f(x) g(x) h(x)
F'(x) = I(x) m(x) n(x) + /,(x) m'(x) n'(x) + I(x) m(x) n(x)
u(x) v(x) w(x) u(x) v(x) w(x) u'(x) v'(x) w'(x)
11. L ' HOPITAL 'S RULE:
(a) Applicable while calculating limits of indeterminate forms of
o 00
the type 0' 00' If the function f(x) and g(x) are differentiable in
certain neighbourhood of the point a, except, may be, at the
point a itself, and g'(x) " 0 , and if
lim f(x)
)1,-+11
= lim g(x) = 0
It ... ;!!
or lim f(x)
It ..... a
= lim g(x) = 00 ,
X -HI
~
, f(x) I' f '(x)
then II m - = Im - - }
,~. g(x) ,~. 9 '(x)
!
J
provided the limit lim f',((X)) exists (L' H opitaI's rule). The point
If. ..... a 9 X i
I
'a' may be either finite or improper + "" or--«>,
(b) Indeterminate forms of the type 0 , "" or "" - "" are reduced to
forms of the type 0o or 0000 by algebraic transformations.
j
(e) Indeterminate forms of the type 1"', ",,0 or 0° are reduced to
forms of the type 0, "" by taking logarithms or by the
transformation If(x)]+(x) _ e+(x)·tnf(x). I
L -__________________________________ ~E
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