# 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)).