Differentiation Formulas Class 12

Here you will learn what is derivative or differentiation and various differentiation formulas class 12.

Let’s begin –

What is Derivative or Differentiation ?

Let f(x) be a differentiable or derivable function on [a, b]. Then,

$lim_{h \to 0}$ $f(x + h) – f(x)\over h$   or,  $lim_{h \to 0}$ $f(x – h) – f(x)\over -h$

is called the derivative or differentiation of f(x) with respect to x and is denoted by

f'(x) or, $d\over dx$ (f(x))  or,  Df(x),  where D = $d\over dx$

Sometimes the derivative or differentiation of the function f(x) is called the differential coefficient of f(x). The process of finding the derivative of a function by using the above definition is called the differentiation from first principles or by ab-initio method or by delta method.

Differentiation Formulas Class 12

Following are deivatives or differentiation of some standard functions.

(i)  $d\over dx$ $x^n$ = $nx^{n-1}$

(ii)  $d\over dx$ $e^x$ = $e^x$

(iii)  $d\over dx$ $a^x$ = $a^xlog_e a$

(iv)  $d\over dx$ (sin x) = cos x

(v)  $d\over dx$ (cos x) = – sin x

(vi)  $d\over dx$ (tan x) = $sec^2 x$

(vii)  $d\over dx$ (cot x) = $- cosec^2 x$

(viii)  $d\over dx$ (sec x) = sec x tan x

(ix)  $d\over dx$ (cosec x) = – cosec x cot x

(x)  $d\over dx$ $sin^{-1} x$ = $1\over {\sqrt{1 – x^2}}$

(xi)  $d\over dx$ $cos^{-1} x$ = – $1\over {\sqrt{1 – x^2}}$

(xii)  $d\over dx$ $tan^{-1} x$ = $1\over {1 + x^2}$

(xiii)  $d\over dx$ $cot^{-1} x$ = -$1\over {1 + x^2}$

(xiv)  $d\over dx$ $sec^{-1} x$ = $1\over {| x |\sqrt{x^2 – 1}}$

(xv)  $d\over dx$ $cosec^{-1} x$ =  – $1\over {| x |\sqrt{x^2 – 1}}$

(xvi)  $d\over dx$ $log_e x$ = $1\over x$

(xvii)  $d\over dx$ $log_a x$ = $1\over xlog_e a$