What is the derivative of (2y/(3x^2+2y^2)) ?

The derivative of (2y/(3x^2+2y^2)) is (2(3x^2\frac{dy)/(dx)-6xy-2y^2(dy)/(dx))}{(3x^2+2y^2)^2}

What is the first derivative of (2y/(3x^2+2y^2)) ?

The first derivative of (2y/(3x^2+2y^2)) is (2(3x^2\frac{dy)/(dx)-6xy-2y^2(dy)/(dx))}{(3x^2+2y^2)^2}

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (3 x^{2} + 2 y^{2}) = 6 x + 4 y \frac{d y}{d x}

= 2 \cdot \frac{\frac{d y}{d x} ( 3 x ^{2} + 2 y ^{2} ) - ( 6 x + 4 y \frac{d y}{d x} ) y}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

Simplify

2 \cdot \frac{\frac{d y}{d x} ( 3 x ^{2} + 2 y ^{2} ) - ( 6 x + 4 y \frac{d y}{d x} ) y}{( 3 x ^{2} + 2 y ^{2} ) ^{2}} : \frac{2 ( 3 x ^{2} \frac{d y}{d x} - 6 x y - 2 y ^{2} \frac{d y}{d x} )}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

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

y^{^{''}} + 14 y^{^{'}} + 85 y = 0

\frac{3 x + 2}{4 x}

x \to 3 lim (x + 3 x - 3)

\int_{- 3}^{0} x 1 - x d x

\int 6 x e^{x} d x

What is the derivative of (2y/(3x^2+2y^2)) ?
The derivative of (2y/(3x^2+2y^2)) is (2(3x^2\frac{dy)/(dx)-6xy-2y^2(dy)/(dx))}{(3x^2+2y^2)^2}
What is the first derivative of (2y/(3x^2+2y^2)) ?
The first derivative of (2y/(3x^2+2y^2)) is (2(3x^2\frac{dy)/(dx)-6xy-2y^2(dy)/(dx))}{(3x^2+2y^2)^2}