(x^(5/2)+2x)/(x^3-4)

Question

What is the derivative of this function, with ALL work.

Answer 1

Given function (x^(5/2)+2x)/(x^3-4)

Aplly derivative with respect of x.

= d/dx[(x^(5/2)+2x)/(x^3-4)]

Recall the formula d/dx(u/v) = (vdu/dx-udv/dx)/v^2

Here u = x^(5/2)+2x,  v = x^3-4

= [(x^3-4)d/dx( x^(5/2)+2x)-(x^(5/2)+2x)d/dx(x^3+4)]/(x^3-4)^2

= [(x^3-4)((5/2x^(5/2)-1)+2)-(x^(5/2)+2x)(3x^2)]/(x^3-4)^2

= [(x^3-4)(5/2x^(3/2)+2)-((x^(5/2)*3x^2)+(2x*3x^2))]/(x^3-4)^2

= [(x^3*5/2x^(3/2))+2x^3-(4*5/2x^(3/2))-8-(3x^(9/2)+2x^5)]/(x^3-4)^2

= [5/2x^(9/2)+2x^3-10x^(3/2)-8-3x^(9/2)-2x^5]/(x^3-4)^2

= [-1/2x^(9/2)-10x^(3/2)+2x^3-2x^5-8]/(x^3-4)^2

Comments

The derivative of the function

Answer 2

Apply derivative with respect of x.

Apply quotient rule in derivatives.