How do you differentiate #f(x)=(3/(5x)-1)/(2/(x^2+7)# using the quotient rule?

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Answer 1 · 2016-12-06T16:58:24

#f'(x) = -(10x^3 - 3x^2 + 21)/(10x^2)#

Let's start by converting this to the from #f(x) = (g(x))/(h(x))#.

#f(x) = ((3 - 5x)/(5x))/(2/(x^2 + 7))#

#f(x) = ((3 - 5x)(x^2 + 7))/(10x)#

#f(x) = (3x^2 - 5x^3 - 35x + 21)/(10x)#

The derivative of a function #f(x) = (g(x))/(h(x))# is given by #color(red)(f'(x) = (g'(x) * h(x) - h'(x)g(x))/(h(x))^2#

#f'(x) = ((-15x^2 + 6x - 35)(10x) - 10(3x^2- 5x^3 - 35x + 21))/(10x)^2#

#f'(x) = (-150x^3 + 60x^2 - 350x - 30x^2 + 50x^3 + 350x -210)/(100x^2)#

#f'(x) = (-100x^3 + 30x^2 - 210)/(100x^2)#

#f'(x) = (-10(10x^3 - 3x^2 + 21))/(100x^2)#

#f'(x) = -(10x^3 - 3x^2 + 21)/(10x^2)#

Hopefully this helps!