Differentiate: f(x) = ((x-7)(x^2+3X)) / x^3
you do this in 2 steps. first differentiate a product (in the paranthesis). then differentiate a difference.
product f(x) = u(x)*v(x), difference: f(x) = u(x)/v(x)
f'(x) = u(x)*v'(x) + v(x)*u'(x) f'(x) = [v(x)*u'(x) - u(x)*v'(x)]/[v(x)^2]
step 1: u(x) = x-7 v(x) = x^2 + 3x
u'(x) = 1 v'(x) = 2x + 3
plug into formula above
f'(x) = (x-7)(2x + 3) + (x^2 + 3x)(1)
f'(x) = 2x^2 + 3x -14x -21 + x^2 + 3x
f'(x) = 3x^2 - 8x - 21
step 2: u(x) = 3x^2 - 8x - 21 v(x) = x^3
u'(x)= 6x - 8 v'(x)= 3x^2
f'(x) = {[x^3 (6x - 8)] - [(3x^2 - 8x - 21)(3x^2)]}/(x^3)^2
f'(x) = {6x^4 -8x^3 - [9x^4 -24x^3 - 63x^2]}/x^6
f'(x) = -3x^4 + 16x^3 - 63x^2/x^6
f'(x) = x^2(-3x^2 + 16x -63)/x^6
f'(x) = -3x^2 + 16x - 63/x^4