What is (\partial)/(\partial x)(1/(sqrt(2x))) ?

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

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(\partial)/(\partial x)(1/(sqrt(2x)))

\frac{\partial}{\partial x} (\frac{1}{2 x})

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

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2 x})

Apply exponent rule:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (2 x) = 2

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

Simplify

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

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

2 x y \frac{d y}{d x} = y^{2} - x^{2} + 1

100 x^{2} + 400 x + 5000, 0.5

\int e^{y (1 - x)} d x

\frac{d}{d x} y = 7 x^{2} cos (x) cot (x)

\frac{d y}{d x} + e^{x} y = - 9 e^{x}

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