What is (\partial)/(\partial x)(x^5+3x^3y^2+3xy^4) ?

The answer to (\partial)/(\partial x)(x^5+3x^3y^2+3xy^4) is 5x^4+9y^2x^2+3y^4

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(\partial)/(\partial x)(x^5+3x^3y^2+3xy^4)

\frac{\partial}{\partial x} (x^{5} + 3 x^{3} y^{2} + 3 x y^{4})

5 x^{4} + 9 y^{2} x^{2} + 3 y^{4}

Solution steps

\frac{\partial}{\partial x} (x^{5} + 3 x^{3} y^{2} + 3 x y^{4})

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x^{5}) + \frac{\partial}{\partial x} (3 x^{3} y^{2}) + \frac{\partial}{\partial x} (3 x y^{4})

\frac{\partial}{\partial x} (x^{5}) = 5 x^{4}

\frac{\partial}{\partial x} (3 x^{3} y^{2}) = 9 y^{2} x^{2}

\frac{\partial}{\partial x} (3 x y^{4}) = 3 y^{4}

= 5 x^{4} + 9 y^{2} x^{2} + 3 y^{4}

What is (\partial)/(\partial x)(x^5+3x^3y^2+3xy^4) ?
The answer to (\partial)/(\partial x)(x^5+3x^3y^2+3xy^4) is 5x^4+9y^2x^2+3y^4