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

The answer to (\partial)/(\partial y)(x/(x^3-y^7)) is (7xy^6)/((x^3-y^7)^2)

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

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

\frac{7 x y ^{6}}{( x ^{3} - y ^{7} ) ^{2}}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (x^{3} - y^{7}) = - 7 y^{6}

= x (- \frac{1}{( x ^{3} - y ^{7} ) ^{2}} (- 7 y^{6}))

Simplify

x (- \frac{1}{( x ^{3} - y ^{7} ) ^{2}} (- 7 y^{6})) : \frac{7 x y ^{6}}{( x ^{3} - y ^{7} ) ^{2}}

= \frac{7 x y ^{6}}{( x ^{3} - y ^{7} ) ^{2}}

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

f (t) = \frac{cos ( t )}{t}

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

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

y^{^{'}} = x y^{3}

What is (\partial)/(\partial y)(x/(x^3-y^7)) ?
The answer to (\partial)/(\partial y)(x/(x^3-y^7)) is (7xy^6)/((x^3-y^7)^2)