What is (\partial)/(\partial y)(x/(x^7-y^3)) ?

The answer to (\partial)/(\partial y)(x/(x^7-y^3)) is (3xy^2)/((x^7-y^3)^2)

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(\partial)/(\partial y)(x/(x^7-y^3))

\frac{\partial}{\partial y} (\frac{x}{x ^{7} - y ^{3}})

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

Solution steps

\frac{\partial}{\partial y} (\frac{x}{x ^{7} - y ^{3}})

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (\frac{1}{x ^{7} - y ^{3}})

Apply exponent rule:

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

= x \frac{\partial}{\partial y} ((x^{7} - y^{3})^{- 1})

Apply the chain rule:

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

= - \frac{1}{( x ^{7} - y ^{3} ) ^{2}} \frac{\partial}{\partial y} (x^{7} - y^{3})

\frac{\partial}{\partial y} (x^{7} - y^{3}) = - 3 y^{2}

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

Simplify

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

= \frac{3 x y ^{2}}{( x ^{7} - y ^{3} ) ^{2}}

\frac{d}{d x} (\frac{3 x ^{2} + 1}{x + 2})

\int \frac{x + 2}{( 3 + 2 x - x ^{2} ) ^{\frac{3}{2}}} d x

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

x \to 2 lim (2 x)

\int \frac{100}{7 x - 16} d x

What is (\partial)/(\partial y)(x/(x^7-y^3)) ?
The answer to (\partial)/(\partial y)(x/(x^7-y^3)) is (3xy^2)/((x^7-y^3)^2)