What is (\partial)/(\partial x)((3x+2xy)/(7x+6y)) ?

The answer to (\partial)/(\partial x)((3x+2xy)/(7x+6y)) is (12y^2+18y)/((7x+6y)^2)

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(\partial)/(\partial x)((3x+2xy)/(7x+6y))

\frac{\partial}{\partial x} (\frac{3 x + 2 x y}{7 x + 6 y})

\frac{12 y ^{2} + 18 y}{( 7 x + 6 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{3 x + 2 x y}{7 x + 6 y})

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

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

\frac{\partial}{\partial x} (7 x + 6 y) = 7

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

Expand

(3 + 2 y) (7 x + 6 y) - 7 (3 x + 2 x y) : 12 y^{2} + 18 y

= \frac{12 y ^{2} + 18 y}{( 7 x + 6 y ) ^{2}}

n = 1 \sum \infty (- \frac{2}{5})^{n}

\frac{1}{( s + 1 ) ( s ^{2} + 1 )}

\frac{d}{d x} (arccoth (x))

\frac{d}{d x} (ln (arctan (5 x^{4})))

\frac{d}{d x} (lo g_{x} (y))

What is (\partial)/(\partial x)((3x+2xy)/(7x+6y)) ?
The answer to (\partial)/(\partial x)((3x+2xy)/(7x+6y)) is (12y^2+18y)/((7x+6y)^2)