# Differentiate root x^2+1/(x-9)^2?

+1 vote
With full working out plz ASAP thanks
asked Jan 30, 2013 in CALCULUS

+1 vote

= √(x2 + 1) / (x - 9)2

Quotient rule: (d/dx)[g(x)/h(x)] =[ h(x) g'(x) - g(x) h'(x)] / h(x)²

={ (x - 9)2 √(x2 + 1)' - √(x2 + 1)[(x - 9)2]'} / [(x - 9)2]2

General derivative formulas: (d/dx)(√x) = 1/(2√x) and (d/dx)(an) = nan-1

= { (x - 9)2 [1/2√(x2 + 1)](2x) - √(x2 + 1)[2(x - 9)](1)} / [(x - 9)2]2

= { (x - 9)2 [1/√(x2 + 1)](x) - √(x2 + 1)[2(x - 9)]} / [(x - 9)2]2

= {[ x(x - 9)2 / √(x2 + 1)] - 2(x - 9)√(x2 + 1) } / [(x - 9)2]2

Take out common factors.

= (x-9){[ x(x - 9) / √(x2 + 1)] - 2√(x2 + 1)} / (x - 9)4

= {[ x(x - 9) / √(x2 + 1)] - 2√(x2 + 1)} / (x - 9)3

Rewrite the expression with common denominator.

= 1/√(x2 + 1){[ x(x - 9) - 2 √(x2 + 1)√(x2 + 1)} / (x - 9)3

= 1/√(x2 + 1){( x2 - 9x) - 2 (x2 + 1)} / (x - 9)3

={( x2 - 9x) - 2 (x2 + 1)} / √(x2 + 1)(x - 9)3

= (x2 - 9x - 2x2 - 2) / [√(x2 + 1)(x - 9)3]

= (- x2 - 9x - 2) / [(x - 9)3√(x2 + 1)]

answered Jan 31, 2013