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Calculus: Differentiation and Applications of Derivatives

A study on the fundamentals of calculus, specifically differentiation and its applications.

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    Calculus: Differentiation and Applications of Derivatives 1. Differentiate the following with respect to ‘x’ a) y = x^4 – 3x^2 + 2 b) y = 3x – 5x^3 c) y = 6 / squareroot x – 4 squareroot x d) y = (x+5) (2x - 1) (Criteria to meet: Differentiate from 1 st principle) Solution: a.

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    ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 2 4 2 4 2 4 2 4 2 3 2 (1) Taking change in and in , 3 2 3 2 3 2 3 2 by (1) y x x x x y y y y x x x x y x x x x y y x x x x x x            = − + + = + − + + = + − + + − = + − + + − − + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 2 2 4 2 2 2 2 4 2 2 2 2 2 2 2 2 4 2 2 4 3 3 2 4 2 3 2 3 2 4 2 3 4 4 3 2 Since 2 2 2 2 2 2 2 2 2 2 4 4 6 4 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x                                + = + + + = + + + + + = + + + + + + + + + = + + + + + + + + + = + + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 4 2 4 3 2 2 4 2 2 3 4 2 4 3 2 2 4 2 Therefore 4 6 4 3 2 2 3 2 4 6 4 3 3 6 2 3 2 y x x x x x x x x x x x x x x y x x x x x x x x x x x x x x               = + + + + − + + + − − + = + + + + − − − + − + − ( ) ( ) ( ) ( ) ( ) ( ) 2 3 4 2 3 2 4 6 4 3 6 y x x x x x x x x x x        = + + + − − ( ) ( ) ( ) ( ) ( ) ( ) 0 2 3 4 2 3 2 0 0 Dividing both sides by and taiking lim 4 6 4 3 6 lim lim x x x x x x x x x x x x x x y x x              → → → + + + − − = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 2 0 0 2 3 3 2 0 0 2 3 3 2 3 4 6 4 3 6 lim lim lim lim 4 6 4 3 6 4 6 0 4 0 0 3 0 6 4 6 x x x x x x x x x x x x x y x x y x x x x x x x x x dy x x x x dx dy x x dx                   → → → → + + + − − = = + + + − − = + + + − − = −

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    b. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 3 3 5 (1) Taking change in and in , 3 5 3 5 3 5 3 5 by (1) y x x x x y y y y x x x x y x x x x y y x x x x x x            = − + = + − + = + − + − = + − + − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 2 3 3 2 3 2 3 3 2 2 3 5 3 3 3 5 3 3 5 5 15 15 3 5 3 5 15 15 y x x x x x x x x x x y x x x x x x x x x x y x x x x x x                = + − + + + − + = + − − − − − + = − − − ( ) ( ) ( ) ( ) ( ) ( ) 0 3 2 2 0 0 2 2 0 0 Dividing both sides by and taiking lim 3 5 15 15 lim lim 3 5 15 15 lim lim x x x x x x x x x x x x y x x x x x x x y x x                    → → → → → − − − = − − − = ( ) ( ) ( ) 2 2 2 3 5 0 15 0 15 3 15 dy x x dx dy x dx = − − − = − c.

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