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

The derivative of \sqrt[3]{(3x/(x+2)}) is 2/(3^{2/3)x^{2/3}(x+2)^{4/3}}

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

The first derivative of \sqrt[3]{(3x/(x+2)}) is 2/(3^{2/3)x^{2/3}(x+2)^{4/3}}

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\frac{2}{3 ^{\frac{2}{3}} x ^{\frac{2}{3}} ( x + 2 ) ^{\frac{4}{3}}}

Solution steps

Apply the chain rule:

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

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

\frac{d}{d x} (\frac{3 x}{x + 2}) = \frac{6}{( x + 2 ) ^{2}}

= \frac{1}{3 ( \frac{3 x}{x + 2} ) ^{\frac{2}{3}}} \cdot \frac{6}{( x + 2 ) ^{2}}

Simplify

\frac{1}{3 ( \frac{3 x}{x + 2} ) ^{\frac{2}{3}}} \cdot \frac{6}{( x + 2 ) ^{2}} : \frac{2}{3 ^{\frac{2}{3}} x ^{\frac{2}{3}} ( x + 2 ) ^{\frac{4}{3}}}

= \frac{2}{3 ^{\frac{2}{3}} x ^{\frac{2}{3}} ( x + 2 ) ^{\frac{4}{3}}}

\int_{0}^{5} π (25 - x^{2}) d x

x \to 4 lim ((x + 3)^{2})

\frac{d ^{2}}{d x ^{2}} (e^{- \frac{x ^{2}}{2}})

\frac{d ^{3}}{d x ^{3}} (sin (x))

What is the derivative of \sqrt[3]{(3x/(x+2)}) ?
The derivative of \sqrt[3]{(3x/(x+2)}) is 2/(3^{2/3)x^{2/3}(x+2)^{4/3}}
What is the first derivative of \sqrt[3]{(3x/(x+2)}) ?
The first derivative of \sqrt[3]{(3x/(x+2)}) is 2/(3^{2/3)x^{2/3}(x+2)^{4/3}}