Summary of Derivatives

Derivatives
Derivative or Differential Coefficient
The rate of change of a quantity y with respect to another quantity x is
called the derivative or differential coefficient of y with respect to x.

Differentiation
The process of finding derivative of a function is called differentiation.

Differentiation using First Principle
Let f ( x ) is a function, differentiable at every point on the real number
line, then its derivative is given by
d f ( x + δx ) − f ( x )
f ′ (x) = f ( x ) = lim
dx δx → 0 δx

Derivatives of Standard Functions
d
(i) ( x n ) = nx n − 1 , n ∈ R
dx
d
(ii) ( k) = 0, where k is constant.
dx
d x
(iii) ( e ) = ex
dx
d
(iv) ( a x ) = a x loge a, where a > 0, a ≠ 1
dx
d 1
(v) (loge x ) = , x > 0
dx x
d 1 1
(vi) (loga x ) = (loga e) = , x> 0
dx x x loge a
d
(vii) (sin x ) = cos x
dx

, d
(viii) (cos x ) = − sin x
dx
d π
(ix) (tan x ) = sec2 x , x ≠ ( 2n + 1) , n ∈ I
dx 2
d
(x) (cot x ) = − cos ec2 x , x ≠ nπ, n ∈ I
dx
d π
(xi) (sec x ) = sec x tan x , x ≠ ( 2n + 1) , n ∈ I
dx 2
d
(xii) (cos ec x ) = − cos ec x cot x , x ≠ n π , n ∈ I
dx
d 1
(xiii) (sin−1 x ) = ,− 1< x < 1
dx 1 − x2
d 1
(xiv) (cos−1 x ) = − , − 1< x < 1
dx 1 − x2
d 1
(xv) (tan−1 x ) =
dx 1 + x2
d 1
(xvi) (cot−1 x ) = −
dx 1 + x2
d 1
(xvii) (sec −1 x ) = ,|x|> 1
dx |x| x 2 − 1
d 1
(xviii) (cos ec−1x ) = − ,|x|> 1
dx |x| x 2 − 1
d
(xix) (sinh x ) = cos h x
dx
d
(xx) (cosh x ) = sin h x
dx
d
(xxi) (tanh x ) = sec h 2x
dx
d
(xxii) (coth x ) = − cos ech 2x
dx
d
(xxiii) (sec h x ) = − sec h x tan h x
dx
d
(xxiv) (cos ech x ) = − cos ech x cot h x
dx

, d
(xxv) (sinh−1 x ) = 1 / ( x 2 + 1)
dx
d
(xxvi) (cosh−1 x ) = 1 / ( x 2 − 1), x > 1
dx
d
(xxvii) (tanh−1 x ) = 1 / (1 − x 2 ),|x|< 1
dx
d
(xxviii) (cot h−1 x ) = 1 / (1 − x 2 ),|x| > 1
dx
d
(xxix) (sec h−1x ) = − 1 / x (1 − x 2 ), x ∈( 0, 1)
dx
d
(xxx) (cos ech−1x ) = − 1/|x| (1 + x 2 ), x ≠ 0
dx

Fundamental Rules for Derivatives
d d
(i) { cf ( x )} = c f ( x ), where c is a constant.
dx dx
d d d
(ii) { f ( x ) ± g( x )} = f(x) ± g( x ) [sum and difference rule]
dx dx dx
d d d
(iii) { f ( x ) g( x )} = f ( x ) g( x ) + g( x ) f(x)
dx dx dx
[leibnitz product rule or product rule]
Generalisation If u1 , u 2 , u3 , ... , u n are functions of x, then
d  du 
( u1 u 2 u3 ... u n ) =  1  [u 2u3 ... u n ]
dx  dx 
 du   du 
+ u1  2  [u3 ... u n ] + u1u 2  3 
 dx   dx 
 du 
[u 4u5K u n ] + K + [u1u 2 ... u n − 1 ]  n 
 dx 
d d
g( x ) f(x) − f(x) g( x )
d  f ( x ) dx dx
(iv)  = [quotient rule]
dx  g( x ) { g( x )} 2
d d
(v) If f ( x ) = φ ( x ), then f ( ax + b) = a φ ( ax + b)
dx dx