Differentiation
1.
DERIVATIVE OF A FUNCTION
Derivative at a Point
The value of f ‘(x) obtained by putting x = a, is called the
 dy 

derivative of f (x) at x = a and it is denoted by f ‘(a) or 
.
 dx  x  a
2.
Standard Derivatives
The following formulae can be applied directly for finding the
derivative of a function:
d
(sin x) = cos x
1.
dx
d
(cos x) = –sin x
dx
3.
d
(tan x) = sec² x
dx
4.
d
(cot x) = –cosec² x
dx
5.
d
(sec x) = sec x tan x
dx
6.
d
(cosec x) = –cosec x cot x
dx
7.
d
(ex) = ex
dx
8.
d
(ax) = ax loge a, a > 1
dx
9.
d
1
(loge x) = , x > 0
dx
x
10.
d
(xn) = nxn – 1
dx
d
d
d
{f (x)} ±
{g(x)}.
{f (x) ± g(x)} =
dx
dx
dx
Note: In general, if f1 (x), f2 (x), … fn (x) are n differentiable functions, then we have
1 x2
, –1 < x < 1
1
1 x
d
[f (x) g(x) h(x)] = f (x) g(x) h’(x) + f (x) g’(x) h(x) + f ’(x) g(x) h(x)
dx
Quotient Rule of Differentiation
If f (x) and g (x) are two differentiable functions of x, then
d
dx


   

f x 
g x f  x  f x g x

 =
.
2
 g x 
g x 


Differentiation of a Function (Chain Rule)
2
, –1 < x < 1
14.
d
(sec–1 x) =
dx
x
17.
3. The derivative of the sum or difference of two function is
the sum or difference of their derivatives, i.e.,
d
[f (x) g(x)] = f (x) g’(x) + g(x) f ‘(x).
dx
1
d
(tan–1 x) =
, –∞ < x < ∞
dx
1 x2
16.
d
d
{c. f (x)} = c
{f (x)}.
dx
dx
If f (x) and g(x) are differentiable functions of x, then
13.
15.
d
(c) = 0.
dx
2. The derivative of constant times a function is constant times
the derivative of the function, i.e.,
1. The derivative of a constant function is zero, i.e.,
PRODUCT RULE OF DIFFERENTIATION
1
d
12.
(cos–1 x) = –
dx
RULES FOR DIFFERENTIATION
d
d
d
d
[f (x) ± f1 (x) ± … ± fn (x)] =
[f1 (x)] ±
[f2 (x)] ± … ±
[fn (x)].
dx 1
dx
dx
dx
2.
d
11.
(sin–1 x) =
dx
3.
1
x2  1
, |x| > 1
−1
d
(cosec–1 x) =
, |x| > 1
dx
x x2 −1
1
d
(cot–1 x) = –
, –∞ < x < ∞
dx
1 x2
x
x
d
(|x|) =
or
, x ≠ 0.
x
dx
x
If y is a differentiable function of u and u is a differentiable
function of x, then
dy
dy
du
=
×
dx
du
dx
Key Points on Chain Rule
1. The chain rule can be extended further as:
If y is a function of u, u is a function of v and v is a function
of x, then
dy
du
dv
dy
=
×
×
and so on.
du
dv
dx
dx
2. If y = un, where u is a function of x, then
dy
dy
du
du
×
= nun – 1 ×
.
=
dx
dx
du
dx
 dy
n1


nu


du


DERIVATIVE OF PARAMETRIC
FUNCTIONS
4.
If x = f (t) and y = g (t), then


f t
dy / dt
dy
=
=
.
dx / dt
g t
dx
And
5.
2
d y
dx
2
=
d  dy 
d  dy 
dt
d  dy  dx
/
=
×
=
.




dx  dx 
dt  dx 
dx
dt  dx  dt
DIFFERENTIATION OF A FUNCTION WITH
RESPECT TO ANOTHER FUNCTION
Step 2. Differentiate w.r.t. x to obtain
dy
dx
Method 2
Step 1. Evaluate
A = Differential coefficient of y treating f (x) as constant.
Step 2. Evaluate
B = Differential coefficient of y treating g(x) as constant.
Step 3.
dy
= A + B.
dx
DIFFERENTIATION OF INVERSE
TRIGONOMETRIC FUNCTIONS
7.
If y = f (x) and z = g(x), then in order to find the derivative of f (x) Important Substitutions to Reduce the Function to
w.r.t. g(x), we use the formula
a Simpler Form


f x
dy / dx
dy
=
=
dz / dx
g x
dz
6.
Expressions Substitutions
a2  x 2 Put x = a sin θ or x = a cos θ
x 2  a2 Put x = a sec θ or x = a cosec θ
LOGARITHMIC DIFFERENTIATION
Properties of Logarithms
1. loge (mn) = loge m + loge n
 m
2.. loge   = loge m – loge n
 n
3. loge (m)n = n loge | m |
4. loge e = 1
5. logn m =
loge m
loge n
6. logn m. logm n = 1.
Shorter Methods of Finding the Derivative of a
Logarithmic Function
dy
, in addition to the method discussed
dx
above, we can also apply any of the following two methods:
If y = [f (x)]g(x), then to find
Method 1
Step 1. Express y = [f (x)]g(x) = eg(x) log f (x)
[ ax = ex log a]
a2  x 2 Put x = a tan θ or x = a cot θ
a x
a x
or
a x
a x
Put x = a tan θ
a x
or
a x
a x
a x
Put x = a cos θ
a2  x 2
a2  x 2
a2  x 2
or
8.
a2  x 2
Put x² = a² cos θ
DIFFERENTIATION OF A FUNCTION
GIVEN IN THE FORM OF A DETERMINANT
  g x  hx 
If y = p  x  q  x  r  x  , then
u x  v x  w x 
f x
( ) g ′ ( x ) h′ ( x ) f ( x ) g ( x ) h ( x ) f ( x ) g ( x ) h ( x )
( ) q (x ) r (x ) + p '(x ) q '(x ) r '(x ) + p (x ) q (x ) r (x )
( ) v (x ) w (x ) u (x ) v (x ) w (x ) u '(x ) v '(x ) w '(x )
f′ x
dy
= p x
dx
u x
Note: The differentiation of a determinant can be done in
columns also.