What is (\partial)/(\partial x)((xy)/(sqrt(xy+4))) ?

The answer to (\partial)/(\partial x)((xy)/(sqrt(xy+4))) is (y(xy+8))/(2(xy+4)sqrt(xy+4))

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(\partial)/(\partial x)((xy)/(sqrt(xy+4)))

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

\frac{y ( x y + 8 )}{2 ( x y + 4 ) x y + 4}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

y \frac{1 \cdot x y + 4 - \frac{y}{2 y x + 4} x}{( x y + 4 ) ^{2}} : \frac{y ( x y + 8 )}{2 ( x y + 4 ) x y + 4}

= \frac{y ( x y + 8 )}{2 ( x y + 4 ) x y + 4}

\frac{3 s e ^{- 3 s}}{4 s ^{2} + 64}

\frac{d}{d x} (x^{\frac{4}{3}} - x^{\frac{1}{3}})

f (x) = (7 - x) (\frac{5}{x ^{2}} - 8)

\frac{d y}{d x}, y = x^{2} + 1

\int e^{6 θ} sin (7 θ) d θ

What is (\partial)/(\partial x)((xy)/(sqrt(xy+4))) ?
The answer to (\partial)/(\partial x)((xy)/(sqrt(xy+4))) is (y(xy+8))/(2(xy+4)sqrt(xy+4))