What is (\partial)/(\partial x)((2x-y)/(5xy+6)) ?

The answer to (\partial)/(\partial x)((2x-y)/(5xy+6)) is (5y^2+12)/((5xy+6)^2)

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(\partial)/(\partial x)((2x-y)/(5xy+6))

\frac{\partial}{\partial x} (\frac{2 x - y}{5 x y + 6})

\frac{5 y ^{2} + 12}{( 5 x y + 6 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{2 x - y}{5 x y + 6})

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

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

\frac{\partial}{\partial x} (5 x y + 6) = 5 y

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

Expand

2 (5 x y + 6) - 5 y (2 x - y) : 5 y^{2} + 12

= \frac{5 y ^{2} + 12}{( 5 x y + 6 ) ^{2}}

\frac{d}{d x} (\frac{7 + 2 ln ( x )}{x})

\int (csc (x) cot (x)) d x

d er i v a t i v e y = e^{x}

x \to \infty lim (6 \cdot x)

\frac{d}{d x} ((1 - x)^{- 1})

What is (\partial)/(\partial x)((2x-y)/(5xy+6)) ?
The answer to (\partial)/(\partial x)((2x-y)/(5xy+6)) is (5y^2+12)/((5xy+6)^2)