What is (\partial)/(\partial x)(sqrt(1+x^2+y^2)) ?

The answer to (\partial)/(\partial x)(sqrt(1+x^2+y^2)) is x/(sqrt(1+x^2+y^2))

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(\partial)/(\partial x)(sqrt(1+x^2+y^2))

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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