What is (\partial)/(\partial x)((x)(x^6y-2x^3y^2)) ?

The answer to (\partial)/(\partial x)((x)(x^6y-2x^3y^2)) is 7x^6y-8x^3y^2

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(\partial)/(\partial x)((x)(x^6y-2x^3y^2))

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

7 x^{6} y - 8 x^{3} y^{2}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x^{6} y - 2 x^{3} y^{2}

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

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

\frac{\partial}{\partial x} (x^{6} y - 2 x^{3} y^{2}) = 6 y x^{5} - 6 y^{2} x^{2}

= 1 \cdot (x^{6} y - 2 x^{3} y^{2}) + (6 y x^{5} - 6 y^{2} x^{2}) x

Simplify

1 \cdot (x^{6} y - 2 x^{3} y^{2}) + (6 y x^{5} - 6 y^{2} x^{2}) x : 7 x^{6} y - 8 x^{3} y^{2}

= 7 x^{6} y - 8 x^{3} y^{2}

\frac{3}{( x ^{2} - 25 ) ^{2}}

2 sin (\frac{π}{4} (x - 5)) + 6

(x^{2} + x + 1)^{5}

ln (z) (z - 3)^{2}

\frac{( 1 + x ^{2} - x ^{3} ) ( 1 - x ^{2} + x ^{3} )}{1 - x ^{2} - x ^{3}}

What is (\partial)/(\partial x)((x)(x^6y-2x^3y^2)) ?
The answer to (\partial)/(\partial x)((x)(x^6y-2x^3y^2)) is 7x^6y-8x^3y^2