What is the (1/(sqrt(x)))^' ?

The (1/(sqrt(x)))^' is -1/(2x^{3/2)}

What is the first (1/(sqrt(x)))^' ?

The first (1/(sqrt(x)))^' is -1/(2x^{3/2)}

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(\frac{1}{x})^{^{'}}

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

Solution steps

(\frac{1}{x})^{'}

Apply exponent rule:

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

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

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

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

Simplify

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

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

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\frac{\partial}{\partial x} (ln (y x))