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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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