DIFFERENTIATION FORMULAS

Differentiation formulas :

Here we are going to see list of formulas used in differentiation.

d  (xn)

d (log x)

d (Constant)

d (√x)

d  (ex)

d  (eax)

d (sin x)

d (sin ax)

d (cos x)

d (cos ax)

d (tan x)

d (tan ax)

d (sec x) 

d (sec ax)

d (cot x)

d (cot ax)

d (cosec x)

d (cosec ax)

d (sin-1 x)

d (cos-1 x)

d (tan-1 x)

d (cosec-1 x)

d (sec-1 x)

d (cot-1 x)

d (ax)

Product rule :

d (uv)

Quotient rule :

d (u/v)

n x (n - 1)

1/x

0

1/2√x

ex

aeax

cos x

a cos ax

-sin x

-a sin ax

sec2x

a sec2ax

sec x tan x

a sec ax tan ax

-cosec2x

-cosec2ax

-cosec x cot x

-a cosec ax cot ax

1/√(1-x2

-1/√(1-x2)

1/(1+x2)

-1/(x√(x2 - 1))

1/(x√(x2 - 1))

-1/(1+x2)

ax log a


u v' + v u'


(vu' - uv')/v2

Differentiation using first principles : 

Formulas in limits :

Example 1 :

Differentiate x⁵ tan x

Solution:

Let y = x⁵ tan x

 u = x⁵         v = tan x

 u' = 5x⁴       v' = sec² x  

(UV)' = UV' + VU'

  =  (x⁵)sec² x  + (tan x)(5x⁴)

  =  x⁵sec² x  + 5x⁴tan x         

  =  x⁴[xsec² x  + tan x]

Example 2 :

Differentiate (x² - 1)/ (x² + 1) with respect to x

Solution :

let y = (x² - 1)/ (x² + 1)

 u = x² - 1                  v = x² + 1

 u' = 2x - 0                 v' = 2x + 0

 u' = 2x                      v' = 2x         

So    y' = [(x² + 1) (2x) - (x² - 1)(2x)] /(x² + 1)²

  =  [(2x)(x² + 1)  - (2x)(x² - 1)] /(x² + 1)²

  =  [(2x³ + 2x)  - (2x³ - 2x)] /(x² + 1)²

  =  [2x³ + 2x  - 2x³ + 2x] /(x² + 1)²

  =  4x /(x² + 1)²

Example 3 :

Differentiate log (sin x) with respect to x

Solution :

 let y = log (sin x) and we are going to take u = sin X

Now the function becomes y = log u 

dy/dx = (dy/du) x (du/dx)

 dy/du = 1/u 

 du/dx = cos X

 dy/dx = (1/u)   x cos X

=  cos X/u

=  cos X/sin X

=   cot X 

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