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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

Simplify

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

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

\int (x) e^{x^{2} - 1} d x

x \to 2 lim (\frac{11 x}{4 - x ^{2}})

(3 x - 5) (x^{2} + 4)

\int \frac{7 x + 14}{x ^{2} + 5 x - 6} d x

\int \frac{1}{3 x ^{2} - 2 x + 4} d x

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