Find the derivative of (sqrt[3]{(3x^{3} + 1}) with respect to x.

Find the derivative of (sqrt[3]{(3x^{3} + 1}) with respect to x.

Correct Answer: Option B

Explanation

(y = sqrt[3]{3x^{3} + 1}  = (3x^{3} + 1)^{frac{1}{3}})

Let u = (3x^{3} + 1); y = (u^{frac{1}{3}})

(frac{mathrm d y}{mathrm d x} = (frac{mathrm d y}{mathrm d u})(frac{mathrm d u}{mathrm d x}))

(frac{mathrm d y}{mathrm d u} = frac{1}{3}u^{frac{-2}{3}})

(frac{mathrm d u}{mathrm d x} = 9x^{2})

(frac{mathrm d y}{mathrm d x} = (frac{1}{3}(3x^{3} + 1)^{frac{-2}{3}})(9x^{2}))

= (frac{3x^{2}}{sqrt[3]{(3x^{3} + 1)^{2}}})