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} )} (- 2 x)

Simplify

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