We're asked to find the derivative
d/(dx) [(-x^2 + 5)/((x^2 + 5)^2)]
Use the quotient rule, which states
d/(dx) [u/v] = (v(du)/(dx) - u(dv)/(dx))/(v^2)
where
u = -x^2 + 5
v = (x^2 + 5)^2:
= ((x^2+5)^2d/(dx)[-x^2+5] - (-x^2+5)d/(dx)[(x^2+5)^2])/(((x^2+5)^2)^2)
Using power rule on first term:
= ((x^2+5)^2(-2x) - (-x^2+5)d/(dx)[(x^2+5)^2])/((x^2+5)^4)
Use the chain rule to differentiate the second term, which in this case is
d/(dx) [(x^2+5)^2] = d/(du)[u^2] (du)/(dx)
where
u = x^2+5
d/(du) [u^2] = 2u (from power rule):
= ((x^2+5)^2(-2x) - (-x^2+5)(2(x^2+5))d/(dx) [x^2+5])/((x^2+5)^4)
= ((x^2+5)^2(-2x) - (-x^2+5)(2x^2+10)(2x))/((x^2+5)^4)
We can simplify this further if we wanted to:
= ((x^2+5)^2(-2x) - (-2)(x^2-5)(x^2+5)(2x))/((x^2+5)^4)
Divide both sides by x^2+5:
= ((x^2+5)(-2x) - (-2)(x^2-5)(2x))/((x^2+5)^3)
= (-2x^3 - 10x - (-2)(x^2-5)(2x))/((x^2+5)^3)
= (-2x^3 - 10x - (20x - 4x^3))/((x^2+5)^3)
= (2x^3 - 30x)/((x^2+5)^3)
= color(blue)((2x(x^2-15))/((x^2+5)^3)
