What is (\partial)/(\partial x)((x^2+y^2+z^2)^{1/2}) ?

The answer to (\partial)/(\partial x)((x^2+y^2+z^2)^{1/2}) is x/((x^2+y^2+z^2)^{1/2)}

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

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

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

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to \infty lim (sec (x))

f (x) = 5 x, at x = 4, x = 9

x \to - 3 - lim (\frac{3 - x}{9 - x ^{2}})

f (x) = 4 x^{\frac{3}{2}}

What is (\partial)/(\partial x)((x^2+y^2+z^2)^{1/2}) ?
The answer to (\partial)/(\partial x)((x^2+y^2+z^2)^{1/2}) is x/((x^2+y^2+z^2)^{1/2)}