What is (\partial)/(\partial x)(1/(sqrt(3x))) ?

The answer to (\partial)/(\partial x)(1/(sqrt(3x))) is -1/(2sqrt(3)x^{3/2)}

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(\partial)/(\partial x)(1/(sqrt(3x)))

\frac{\partial}{\partial x} (\frac{1}{3 x})

- \frac{1}{2 3 x ^{\frac{3}{2}}}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{3 x})

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{1}{2 ( 3 x ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (3 x)

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

\frac{\partial}{\partial x} (3 x) = 3

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

Simplify

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

= - \frac{1}{2 3 x ^{\frac{3}{2}}}

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What is (\partial)/(\partial x)(1/(sqrt(3x))) ?
The answer to (\partial)/(\partial x)(1/(sqrt(3x))) is -1/(2sqrt(3)x^{3/2)}