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

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

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(\partial)/(\partial y)(sqrt(x^2+y^2+z^2))

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

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

Solution steps

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

Treat

x, z

as constants

Apply the chain rule:

\frac{1}{2 x ^{2} + y ^{2} + z ^{2}} \frac{\partial}{\partial y} (x^{2} + y^{2} + z^{2})

= \frac{1}{2 x ^{2} + y ^{2} + z ^{2}} \frac{\partial}{\partial y} (x^{2} + y^{2} + z^{2})

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

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

Simplify

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

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

\frac{d}{d x} (\frac{1}{2} x + 1)

\frac{\partial}{\partial x} (cos (x + 4 y))

200 + \frac{7}{x} + \frac{x ^{2}}{7}

\int \frac{- 4 x + 1}{x ^{2} + 2} d x

\frac{d}{d x} (\frac{9}{( 2 x + 9 ) ^{2}})

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