Mathematics

Session Differentiation - 2

Session Objectives  Fundamental Rules, Product Rule and Quotient Rule  Differentiation of Function of a Function  Differentiation by Trigonometric Substitutions  Differentiation of Implicit Functions  Class Exercise

Fundamental Rules

Example-1 Differentiate the following:

Solution Differentiating y with respect to x, we get

Product Rule Ifand are differentiable functions, then

Example-2 Differentiating w.r.t. x, we get Differentiate: w.r.t. x. Solution Let y =x 2 sinxlogx

Quotient Rule Ifand are differentiable functions and, then

Example-3 Differentiate: w.r.t. x. Solution: Differentiating w.r.t. x, we get

Differentiation of Function of a Function Note: If and, then If ƒ(x) and g(x) are differentiable functions, then ƒog is also differentiable (Chain Rule)

Example-4 Differentiatew. r. t. x. Solution: Differentiating y w.r.t. x, we get

Example-5

Continued

Trigonometric Substitutions

Example-6 Putting x 2 = cos2 Solution:

Continued Differentiating y w.r.t. x, we get

Example-7

Continued

Differentiation of Implicit Functions y is not expressible directly in terms of x

Example-8 We have xy 3 – yx 3 = x Solution:

Solution Cont.

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