DERIVATIVE OF RATIONAL FUNCTION WITHOUT USING QUOTIENT RULE

Here we see derivatives of rational functions without using quotient rule.

Problem 1 :

y = 2x + 5 / x

Solution :

y = 2x+5xy = 2xx + 5x-1y = 2 + 5x-1dydx = 0 + (-1)5x-1-1dydx = -5x-2dydx = -5x2

Problem 2 :

y = (x³ - 2) / x

Solution :

y = x3-2xy = x3x - 2x-1y = x2 - 2x-1dydx = 2x - (-1)2x-1-1dydx = 2x+2x-2dydx = 2x+2x2

Problem 3 :

y = (x³ - 2) / x

Solution :

y = x2-4x+7xy = x2x -4xx + 7xy = x-4+7x-1dydx = 1-0+7(-1) x-1-1dydx = 1-7x-2dydx = 1-7x2

Problem 4 :

y = (x3 - 4x2 + 3x - 2)/x2

Solution :

y = x3-4x2+3x-2x2y = x3x2 -4x2x2+3xx2-2x2y = x-4+3x-2x2y = x-4+3x-1-dydx = 1-0+3(-1) x-1-1-dydx = 1-3 x-2+dydx = 1-3x24x3

Problem 5 :

y = (3x7 - 7x + 11)/2x3

Solution :

y = 3x7-7x+112x3y = 3x72x3 -7x2x3+112x3y = 3x42-72x2+112x3y = 3x42-7x-22+11x-32dydx = 34x32-7(-2)x-2-12+11(-3)x-3-12dydx = 6x3+7 x-3-33x-42dydx = 6x3+7x3-332x4

Problem 6 :

y = (2x + 3)(2x - 3)/x

Solution :

y = ((2x)2 - 32)/x

y = (4x2 - 9)/x

y = 4x2-9xy = 4x2x9xy = 4x-9x-1= 4(1)-9(-1)x-1-1= 4+9x-2= 4+9x2

Problem 7 :

y = (x + 6)/x3

Solution :

y =x+6x3y = xx3+6x3y =1x2+6x3y =x-2+6x-3 dydx= -2x-2-1+6(-3)x-3-1dydx= -2x3-18x4

Problem 8 :

y = (2x3 + x + 4)/2x5

Solution :

y =2x3+x+42x5y = 2x32x5+x2x542x5y =1x2+12x52x5y =x-2+12x-5+2x-5 dydx= -2x-2-1+12(-5)x-5-1+2(-5)x-5-1dydx= -2x-3-52x-6-10x-6dydx= -2x3-52x6-10x6

Problem 9 :

y = (x - 3)/√x

Solution :

y =x-3xy = xx-3xy = xx-1/2- 3x-1/2y = x1/2- 3x-1/2dydx =12 x1/2 - 1 - 3-12x-1/2-1dydx =12 x-1/2 + 32x-3/2dydx =12x + 32xx

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