What is (\partial)/(\partial x)(xsqrt(x^2+y^2)) ?

The answer to (\partial)/(\partial x)(xsqrt(x^2+y^2)) is (2x^2+y^2)/(sqrt(x^2+y^2))

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(\partial)/(\partial x)(xsqrt(x^2+y^2))

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

\frac{2 x ^{2} + y ^{2}}{x ^{2} + y ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = x^{2} + y^{2}

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

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

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

= 1 \cdot x^{2} + y^{2} + \frac{x}{x ^{2} + y ^{2}} x

Simplify

1 \cdot x^{2} + y^{2} + \frac{x}{x ^{2} + y ^{2}} x : \frac{2 x ^{2} + y ^{2}}{x ^{2} + y ^{2}}

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

y = 3 x, y = \frac{3}{x}

\frac{\partial}{\partial z} (4 x^{\frac{y}{z}})

2 x - 10, [4, 8]

cos (\frac{3}{2} x)

2000 (1 - \frac{x}{600})^{2}

What is (\partial)/(\partial x)(xsqrt(x^2+y^2)) ?
The answer to (\partial)/(\partial x)(xsqrt(x^2+y^2)) is (2x^2+y^2)/(sqrt(x^2+y^2))