y = 4x / (x^2 + 5). Find dy/dx.

We use the quotient rule here which states that if y = f(x)/g(x) then dy/dx = (f'(x)g(x) - g'(x)f(x)) / (g(x)^2). Here f(x) = 4x and g(x) = x^2 + 5, so we have f'(x) = 4 , g'(x) = 2x. This gives us dy/dx = (4(x^2 + 5) - 2x(4x)) / ((x^2 + 5)^2) = (4x^2 + 20 - 8x^2) / ((x^2 + 5)^2) = (20 - 4x^2) / ((x^2 + 5)^2).