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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

= π (\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}}

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

Simplify

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

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

\frac{d}{d x} (ln (\frac{1}{6 + x}))

x \to 2 lim (x + 1)

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

\int 4 x^{3} + 9 sin (x) d x

x \to 3 lim (x + 1)

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