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

The answer to (\partial)/(\partial x)((3x+2y)/(y^2+5x^2)) is (3y^2-15x^2-20xy)/((y^2+5x^2)^2)

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

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

\frac{3 y ^{2} - 15 x ^{2} - 20 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

Solution steps

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

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

\frac{\partial}{\partial x} (y^{2} + 5 x^{2}) = 10 x

= \frac{3 ( y ^{2} + 5 x ^{2} ) - 10 x ( 3 x + 2 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

Expand

3 (y^{2} + 5 x^{2}) - 10 x (3 x + 2 y) : 3 y^{2} - 15 x^{2} - 20 x y

= \frac{3 y ^{2} - 15 x ^{2} - 20 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

\int 3 + 4 x d x

\int \frac{1}{x ^{4} 4 - x ^{2}} d x

\int \frac{e ^{5 x}}{5} d x

\int_{0}^{1} \frac{2 x}{( x ^{2} + 3 ) ^{2}} d x

f (x) = \frac{- 9 x ^{2}}{2} + 7 x + 2

What is (\partial)/(\partial x)((3x+2y)/(y^2+5x^2)) ?
The answer to (\partial)/(\partial x)((3x+2y)/(y^2+5x^2)) is (3y^2-15x^2-20xy)/((y^2+5x^2)^2)