Differentiation Formulas Class 12 – Calculus

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 derivatives or differentiation of some standard functions.

Basic Differentiation Formulas

(i)  \(d\over dx\) \(x^n\) = \(nx^{n-1}\)

(ii)  \(d\over dx\)(a) = 0,     where a is constant.

(iii)   \(d\over dx\)(x) = 1

(iv)  \(d\over dx\)(kx) = k,    where k is constant

Differentiation of Logarithmic and Exponential Function Formulas

(i)   \(d\over dx\) \(e^x\) = \(e^x\)

(ii)  \(d\over dx\) \(a^x\) = \(a^xlog_e a\)

(iii)  \(d\over dx\) \(log_e x\) = \(1\over x\)

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

Trigonometric Function Differentiation Formulas Class 12

(i)  \(d\over dx\) (sin x) = cos x

(ii)  \(d\over dx\) (cos x) = – sin x

(iii)  \(d\over dx\) (tan x) = \(sec^2 x\)

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

(vi)  \(d\over dx\) (sec x) = sec x tan x

(vi)  \(d\over dx\) (cosec x) = – cosec x cot x

Inverse Trigonometric Function Differentiation Formulas

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

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

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

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

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

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

Differentiation Rules Class 12

(i)  Product Rule – \(d\over dx\) {f(x) g(x)} = \(d\over dx\) (f(x)) g(x) + f(x). \(d\over dx\) (g(x))

(ii)  Quotient Rule – \({g(x) {d\over dx} (f(x)) – f(x) {d\over dx} (g(x))}\over {(g(x))^2}\)

(iii)  Chain Rule – \(d\over dx\) {(fog) (x)} = \(d\over d g(x)\) {(fog) (x)} \(d\over dx\) (g(x)).

Leave a Comment

Your email address will not be published. Required fields are marked *