What is the d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) ?

The d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) is ({x})/({x)^2+{y}^2+{z}^2}

What is the first d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) ?

The first d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) is ({x})/({x)^2+{y}^2+{z}^2}

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is the d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) ?
The d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) is ({x})/({x)^2+{y}^2+{z}^2}
What is the first d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) ?
The first d/(d{x)}(ln(sqrt({x)^2+{y}^2+{z}^2})) is ({x})/({x)^2+{y}^2+{z}^2}