What is expand (3x^2+3y^2(dy)/(dx))(x^2+y^2) ?

The solution to expand (3x^2+3y^2(dy)/(dx))(x^2+y^2) is 3x^4+3x^2y^2+3x^2y^2(dy)/(dx)+3y^4(dy)/(dx)

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expand (3x^2+3y^2(dy)/(dx))(x^2+y^2)

expand

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

3 x^{4} + 3 x^{2} y^{2} + 3 x^{2} y^{2} \frac{d y}{d x} + 3 y^{4} \frac{d y}{d x}

Solution steps

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

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 3 x^{2}, b = 3 y^{2} \frac{d y}{d x}, c = x^{2}, d = y^{2}

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

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

Simplify

3 x^{2} x^{2} + 3 x^{2} y^{2} + 3 x^{2} y^{2} \frac{d y}{d x} + 3 y^{2} y^{2} \frac{d y}{d x} : 3 x^{4} + 3 x^{2} y^{2} + 3 x^{2} y^{2} \frac{d y}{d x} + 3 y^{4} \frac{d y}{d x}

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

\frac{2 ( 2 n + 3 ) \cdot ( 5 n + 3 )}{3 ( 5 n + 8 ) \cdot ( 2 n + 1 )}

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x + 8 x + 16

(cos (x) + a sin (x))^{2}

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

What is expand (3x^2+3y^2(dy)/(dx))(x^2+y^2) ?
The solution to expand (3x^2+3y^2(dy)/(dx))(x^2+y^2) is 3x^4+3x^2y^2+3x^2y^2(dy)/(dx)+3y^4(dy)/(dx)