What is (\partial)/(\partial x)((y/x)^{3/4}) ?

The answer to (\partial)/(\partial x)((y/x)^{3/4}) is -(3y^{3/4})/(4x^{7/4)}

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(\partial)/(\partial x)((y/x)^{3/4})

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{3}{4}})

- \frac{3 y ^{\frac{3}{4}}}{4 x ^{\frac{7}{4}}}

Solution steps

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{3}{4}})

Treat

y

as a constant

Apply the chain rule:

\frac{3}{4 ( \frac{y}{x} ) ^{\frac{1}{4}}} \frac{\partial}{\partial x} (\frac{y}{x})

= \frac{3}{4 ( \frac{y}{x} ) ^{\frac{1}{4}}} \frac{\partial}{\partial x} (\frac{y}{x})

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

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

Simplify

\frac{3}{4 ( \frac{y}{x} ) ^{\frac{1}{4}}} (- \frac{y}{x ^{2}}) : - \frac{3 y ^{\frac{3}{4}}}{4 x ^{\frac{7}{4}}}

= - \frac{3 y ^{\frac{3}{4}}}{4 x ^{\frac{7}{4}}}

\frac{\partial}{\partial y} (e^{x} cos (2 y))

y^{^{'}} + 4 y = 32 x, y (0) = 3

\frac{d y}{d x} = x y + 5 x + 4 y + 20

\frac{d}{d x} (7 e^{- 5 x})

\frac{d}{d x} (\frac{10}{e ^{x}})

What is (\partial)/(\partial x)((y/x)^{3/4}) ?
The answer to (\partial)/(\partial x)((y/x)^{3/4}) is -(3y^{3/4})/(4x^{7/4)}