# **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**

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

**PRODUCT RULE OF DIFFERENTIATION**

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

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

**Solution**

**QUOTIENT RULE OF DIFFERENTIATION**

** (OR)**

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**Example: Differentiate , w. r. t. x****Solution: **

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**Chain Rule of finding the differentiations**

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

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

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## 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**

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**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**

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**⇒ 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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