Need help finding derivative.?

Question

x^3 - 1 / (3x-1)^2.

I would prefer it if you used the quotient rule to solve.

Answer 1

d /dx [X³- 1 / (3x - 1)²]

Put X³-1 = f(x) and (3x -1)² = g(x)

Derivative  each side with  respective x

Therefore 3x2dx = f¹(x)dx  and 2(3x - 1)3dx = g¹(x)dx

Simplify

f¹(x)dx= 3x2dx  and g¹(x)dx = 6dx ( 3x - 1)

Recall : d / dx [f(x)/g(x)] = [g(x)f¹(x) - f(x)g¹(x)] / [g(x)]²

Substitue

d / dx [x³ - 1 / (3x - 1)²]  = (3x - 1)²[3x²] - [(x³ - 1)6(3x -1)] / [(3x -1)²]²

Simplify

d / dx [x³ - 1/ (3x - 1)²] = (3x - 1) [ (3x - 1)3x² - 6(x³ - 1) ] / (3x - 1)⁴

Simplify

d / dx [x³ - 1/ (3x - 1)²] = [ (3x - 1)3x² - (x³ - 1)6 ] / (3x - 1)³

 Derivative of [ x³ - 1 / (3x - 1)²] = [ 3(3x - 1)x² - 6(x³ - 1) ] / (3x - 1)³