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Find the derivative of $\sqrt[3]{(3x^{3} + 1}$ with respec...

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

A.

$\frac{3x}{3(3x^{3} + 1)}$

B.

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

C.

$\frac{3x}{\sqrt[3]{3x^{2} + 1}}$

D.

$\frac{3x^{2}}{3(3x^{2} + 1)^{2}}$

View answer & explanation

Correct answer is B

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

Let u = $3x^{3} + 1$ ; y = $u^{\frac{1}{3}}$

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

$\frac{\mathrm d y}{\mathrm d u} = \frac{1}{3}u^{\frac{-2}{3}}$

$\frac{\mathrm d u}{\mathrm d x} = 9x^{2}$

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

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

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