What is (\partial)/(\partial x)((5x-y)/(y^2+3x^2)) ?

The answer to (\partial)/(\partial x)((5x-y)/(y^2+3x^2)) is (5y^2-15x^2+6xy)/((y^2+3x^2)^2)

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(\partial)/(\partial x)((5x-y)/(y^2+3x^2))

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

\frac{5 y ^{2} - 15 x ^{2} + 6 x y}{( y ^{2} + 3 x ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( 5 x - y ) ( y ^{2} + 3 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 3 x ^{2} ) ( 5 x - y )}{( y ^{2} + 3 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (5 x - y) = 5

\frac{\partial}{\partial x} (y^{2} + 3 x^{2}) = 6 x

= \frac{5 ( y ^{2} + 3 x ^{2} ) - 6 x ( 5 x - y )}{( y ^{2} + 3 x ^{2} ) ^{2}}

Expand

5 (y^{2} + 3 x^{2}) - 6 x (5 x - y) : 5 y^{2} - 15 x^{2} + 6 x y

= \frac{5 y ^{2} - 15 x ^{2} + 6 x y}{( y ^{2} + 3 x ^{2} ) ^{2}}

y = (x^{3} + 1)^{- 1}, x = 0, x = 1

\int x^{2} + 4 x + 4 d x

\int_{0}^{4} - x + 1 d x

x \to \infty lim (\frac{x}{6 + e ^{x}})

\int (\frac{4 - t + t ^{2}}{t ^{2}}) d t

What is (\partial)/(\partial x)((5x-y)/(y^2+3x^2)) ?
The answer to (\partial)/(\partial x)((5x-y)/(y^2+3x^2)) is (5y^2-15x^2+6xy)/((y^2+3x^2)^2)