What is (\partial)/(\partial x)(xln(z^4+x^2)) ?

The answer to (\partial)/(\partial x)(xln(z^4+x^2)) is ln(z^4+x^2)+(2x^2)/(z^4+x^2)

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(\partial)/(\partial x)(xln(z^4+x^2))

\frac{\partial}{\partial x} (x ln (z^{4} + x^{2}))

ln (z^{4} + x^{2}) + \frac{2 x ^{2}}{z ^{4} + x ^{2}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{4} + x^{2}))

Treat

z

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = ln (z^{4} + x^{2})

= \frac{\partial}{\partial x} (x) ln (z^{4} + x^{2}) + \frac{\partial}{\partial x} (ln (z^{4} + x^{2})) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (ln (z^{4} + x^{2})) = \frac{2 x}{z ^{4} + x ^{2}}

= 1 \cdot ln (z^{4} + x^{2}) + \frac{2 x}{z ^{4} + x ^{2}} x

Simplify

1 \cdot ln (z^{4} + x^{2}) + \frac{2 x}{z ^{4} + x ^{2}} x : ln (z^{4} + x^{2}) + \frac{2 x ^{2}}{z ^{4} + x ^{2}}

= ln (z^{4} + x^{2}) + \frac{2 x ^{2}}{z ^{4} + x ^{2}}

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What is (\partial)/(\partial x)(xln(z^4+x^2)) ?
The answer to (\partial)/(\partial x)(xln(z^4+x^2)) is ln(z^4+x^2)+(2x^2)/(z^4+x^2)