How do you differentiate $\frac{x^{2} + x + 3}{\sqrt{x - 3}}$ $(x^2 + x + 3 )/ sqrt(x-3)$ using the quotient rule?

Question

1951 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-11-27T20:56:02

$h'(x)= -[3(x+1)]/((x-3)^(3/2))$

The quotient rule; given $f(x)!=0$

if $h(x)= f(x)/g(x)$ ; then $h'(x)= [g(x)*f'(x) -f(x)*g'(x)]/(g(x))^2$

given $h(x)= (x^2+x+3)/root()(x-3)$

let $f(x)= x^2 +x+3$
$color (red) (f'(x)= 2x+1)$

let $g(x)= root()(x-3) = (x-3)^(1/2)$

$color(blue)(g'(x)= 1/2 (x-3)^(1/2-1)= 1/2 (x-3)^(-1/2)$

$h'(x)= [(x-3)^(1/2)*color(red)((2x+1))-color(blue)(1/2(x-3)^(-1/2))(x^2 +x+3)]/(root()[(x-3)]^2$

Factor out the greatest common factor $1/2 (x-3)^(-1/2)$

$h'(x)= 1/2 (x-3)^(-1/2)[(x-3)(2x+1)-(x^2 +x+3)]/(x-3)$

$=>h'(x) = 1/2[(x^2 +x-6x-3-x^2 -x-3)]/(x-3)^(3/2)$
$h'(x)= (-6x-6)/(2(x-3)^(3/2))$

$h'(x)= -[6(x+1)]/(2(x-3)^(3/2))$

$color(red)(h'(x)= -[3(x+1)]/((x-3)^(3/2)))$ Answer

How do you differentiate (x^2 + x + 3 )/ sqrt(x-3)x2+x+3√x−3(x^2 + x + 3 )/ sqrt(x-3) using the quotient rule?