What is (\partial)/(\partial x)((3xy)/(x^4+y^4)) ?

The answer to (\partial)/(\partial x)((3xy)/(x^4+y^4)) is (3y(-3x^4+y^4))/((x^4+y^4)^2)

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(\partial)/(\partial x)((3xy)/(x^4+y^4))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

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

h \to 0 lim (\frac{( 2 + h ) ^{2} - 8}{h})

\frac{\partial}{\partial x} (ln (x^{2}))

\frac{\partial}{\partial z} (x z - 5 x^{2} y^{3} z^{4})

\int (\frac{3}{x} - \frac{5}{x ^{6}}) d x

What is (\partial)/(\partial x)((3xy)/(x^4+y^4)) ?
The answer to (\partial)/(\partial x)((3xy)/(x^4+y^4)) is (3y(-3x^4+y^4))/((x^4+y^4)^2)