# **Differentiation For Classes 11 & 12**

Differentiation formulas and basic concepts for classes 11 and 12 strictly according to the CBSE syllabus. Basic formulas of calculus

## Differentiation formulas

**DERIVATIVE BY FIRST PRINCIPAL**

## **DIFFERENTIATION OF SOME IMPORTANT FUNCTIONS**

**DIFFERENTIATION OF SOME TRIGONOMETRIC FUNCTION****S**

**PRODUCT RULE OF DIFFERENTIATION**

(uv)' = u'v + uv' OR

Example : Differentiate y = x^{2}sinx, w.r.t. x

Solution

**QUOTIENT RULE OF DIFFERENTIATION**

(OR)

Example: Differentiate

, w. r. t. x

Solution:

**Chain Rule of finding the differentiations**

Example: Differentiate f(x) = (sin3x)^{4} with respect to x

Solution: f(x) = (sin3x)^{4}

## CONCEPT OF DIFFERENTIABILITY

A function is formally considered differentiable if its derivative exists at each point in its domain.

For a function to be differentiable it nust be continuous.

**DEFINITION OF DIFFERENTIABILITY**

f(x) is said to be differentiable at the point x = a if the derivative f ‘(a) exists at every point in its domain. It is given by

This formula is derived from Lagranges Mean Value Theorem.

Putting x = a + h, as x → a, h → 0 so this can be written as

**CALCULATING DIFFERENTIABILITY**

For checking the differentiability of a function we have to calculate Left Hand Differentiability (LHD) and Right Hand Differentiability (RHD)

It is batter to use basic formula of finding the differentiability as given below

If LHD = RHD then the function is differentiable

**Note:** If a function is differentiable at any point, it is necessarily continuous at that point.

Example: Examine the function for differentiability at x = 2. Solution:

At x < 2, f(x) = 1 + x

At x > 2 f(x) = 5 - x

⇒ At x = 2, Lf ' (2) ≠ Rf ' (2)

⇒ LHD ≠ RHD

⇒ f(x) is not differentiable at x = 2

**DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS**

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