What is the derivative of \sqrt[3]{2+2x+x^3} ?

The derivative of \sqrt[3]{2+2x+x^3} is (3x^2+2)/(3(2+2x+x^3)^{2/3)}

What is the first derivative of \sqrt[3]{2+2x+x^3} ?

The first derivative of \sqrt[3]{2+2x+x^3} is (3x^2+2)/(3(2+2x+x^3)^{2/3)}

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derivative of

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

Solution steps

Apply the chain rule:

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

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

\frac{d}{d x} (2 + 2 x + x^{3}) = 3 x^{2} + 2

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

Simplify

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

= \frac{3 x ^{2} + 2}{3 ( 2 + 2 x + x ^{3} ) ^{\frac{2}{3}}}

What is the derivative of \sqrt[3]{2+2x+x^3} ?
The derivative of \sqrt[3]{2+2x+x^3} is (3x^2+2)/(3(2+2x+x^3)^{2/3)}
What is the first derivative of \sqrt[3]{2+2x+x^3} ?
The first derivative of \sqrt[3]{2+2x+x^3} is (3x^2+2)/(3(2+2x+x^3)^{2/3)}