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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

= (y + 3) \frac{1 \cdot ( x ^{2} + y ) - 2 xx}{( x ^{2} + y ) ^{2}}

Simplify

(y + 3) \frac{1 \cdot ( x ^{2} + y ) - 2 xx}{( x ^{2} + y ) ^{2}} : \frac{( - x ^{2} + y ) ( y + 3 )}{( x ^{2} + y ) ^{2}}

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

x^{4} e^{y} + (x + 1) y \frac{d y}{d x} = 0

f (x) = (1 + cos^{2} (x))^{6}

arcsin (x)

\int_{5}^{10} a x - 10 b d x

f (x) = \frac{12}{x}, at x = - 2

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