What is (\partial)/(\partial x)(1/(2pi)(y/x)^{1/2}) ?

The answer to (\partial)/(\partial x)(1/(2pi)(y/x)^{1/2}) is -(y^{1/2})/(4pix^{3/2)}

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(\partial)/(\partial x)(1/(2pi)(y/x)^{1/2})

\frac{\partial}{\partial x} (\frac{1}{2 π} (\frac{y}{x})^{\frac{1}{2}})

- \frac{y ^{\frac{1}{2}}}{4 π x ^{\frac{3}{2}}}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2 π} (\frac{y}{x})^{\frac{1}{2}})

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{2 π} \cdot \frac{1}{2 ( \frac{y}{x} ) ^{\frac{1}{2}}} (- \frac{y}{x ^{2}}) : - \frac{y ^{\frac{1}{2}}}{4 π x ^{\frac{3}{2}}}

= - \frac{y ^{\frac{1}{2}}}{4 π x ^{\frac{3}{2}}}

\frac{d}{d x} (a e^{3 x})

s (t) = \frac{1}{5} t^{4}

(7 - 3 x^{2})^{3}

\frac{d}{d x} (\frac{x + 36}{x})

\frac{\partial}{\partial x} (6 e^{y} sin (4 x) + 3 y - 2 x y)

What is (\partial)/(\partial x)(1/(2pi)(y/x)^{1/2}) ?
The answer to (\partial)/(\partial x)(1/(2pi)(y/x)^{1/2}) is -(y^{1/2})/(4pix^{3/2)}