What is the derivative of 1/(3(x+2^{2/3)}) ?

The derivative of 1/(3(x+2^{2/3)}) is -2/(9(x+2)^{5/3)}

What is the first derivative of 1/(3(x+2^{2/3)}) ?

The first derivative of 1/(3(x+2^{2/3)}) is -2/(9(x+2)^{5/3)}

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

- \frac{2}{9 ( x + 2 ) ^{\frac{5}{3}}}

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{2}{3 ( x + 2 ) ^{\frac{5}{3}}} \frac{d}{d x} (x + 2)

= - \frac{2}{3 ( x + 2 ) ^{\frac{5}{3}}} \frac{d}{d x} (x + 2)

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

= \frac{1}{3} (- \frac{2}{3 ( x + 2 ) ^{\frac{5}{3}}} \cdot 1)

Simplify

\frac{1}{3} (- \frac{2}{3 ( x + 2 ) ^{\frac{5}{3}}} \cdot 1) : - \frac{2}{9 ( x + 2 ) ^{\frac{5}{3}}}

= - \frac{2}{9 ( x + 2 ) ^{\frac{5}{3}}}

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What is the derivative of 1/(3(x+2^{2/3)}) ?
The derivative of 1/(3(x+2^{2/3)}) is -2/(9(x+2)^{5/3)}
What is the first derivative of 1/(3(x+2^{2/3)}) ?
The first derivative of 1/(3(x+2^{2/3)}) is -2/(9(x+2)^{5/3)}