What is the derivative of-y/(x+y^2) ?

The derivative of-y/(x+y^2) is -(x\frac{dy)/(dx)-y^2(dy)/(dx)-y}{(x+y^2)^2}

What is the first derivative of-y/(x+y^2) ?

The first derivative of-y/(x+y^2) is -(x\frac{dy)/(dx)-y^2(dy)/(dx)-y}{(x+y^2)^2}

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

Expand

\frac{d y}{d x} (x + y^{2}) - (1 + 2 y \frac{d y}{d x}) y : x \frac{d y}{d x} - y^{2} \frac{d y}{d x} - y

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

\int \frac{sin ( t )}{cos ^{2} ( t )} d t

\int \frac{29}{x ^{3} - 27} d x

\frac{d y}{d x} + y cos (x) = 7 cos (x)

\frac{\partial}{\partial x} (x^{2} + 4 x y + y^{6} - 8)

\frac{d}{d x} (4 x^{2} e^{2 x^{2} + x - 1})

What is the derivative of-y/(x+y^2) ?
The derivative of-y/(x+y^2) is -(x\frac{dy)/(dx)-y^2(dy)/(dx)-y}{(x+y^2)^2}
What is the first derivative of-y/(x+y^2) ?
The first derivative of-y/(x+y^2) is -(x\frac{dy)/(dx)-y^2(dy)/(dx)-y}{(x+y^2)^2}