## 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/x01/2√xexaeaxcos xa cos ax-sin x-a sin axsec2xa sec2axsec x tan xa sec ax tan ax-cosec2x-cosec2ax-cosec x cot x-a cosec ax cot ax1/√(1-x2) -1/√(1-x2)1/(1+x2)-1/(x√(x2 - 1))1/(x√(x2 - 1))-1/(1+x2)ax log au 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

After having gone through the stuff given above, we hope that the students would have understood "Differentiation formulas".

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