What is (\partial)/(\partial x)(xln(z^3+x^3)) ?

The answer to (\partial)/(\partial x)(xln(z^3+x^3)) is ln(z^3+x^3)+(3x^3)/(z^3+x^3)

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(\partial)/(\partial x)(xln(z^3+x^3))

\frac{\partial}{\partial x} (x ln (z^{3} + x^{3}))

ln (z^{3} + x^{3}) + \frac{3 x ^{3}}{z ^{3} + x ^{3}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{3} + x^{3}))

Treat

z

as a constant

Apply the Product Rule:

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

f = x, g = ln (z^{3} + x^{3})

= \frac{\partial}{\partial x} (x) ln (z^{3} + x^{3}) + \frac{\partial}{\partial x} (ln (z^{3} + x^{3})) x

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

\frac{\partial}{\partial x} (ln (z^{3} + x^{3})) = \frac{3 x ^{2}}{z ^{3} + x ^{3}}

= 1 \cdot ln (z^{3} + x^{3}) + \frac{3 x ^{2}}{z ^{3} + x ^{3}} x

Simplify

1 \cdot ln (z^{3} + x^{3}) + \frac{3 x ^{2}}{z ^{3} + x ^{3}} x : ln (z^{3} + x^{3}) + \frac{3 x ^{3}}{z ^{3} + x ^{3}}

= ln (z^{3} + x^{3}) + \frac{3 x ^{3}}{z ^{3} + x ^{3}}

\int 9 1 + e^{x} d x

y^{^{''}} + 2 y^{^{'}} + 2 y = e^{- t}, y (0) = 0, y^{^{'}} (0) = 1

\frac{d}{d x} (x^{2} - 3 x + 5)

\frac{d}{d x} (- \frac{11}{x ^{6}})

f (x) = 3 x - 4 x^{\frac{9}{10}}

What is (\partial)/(\partial x)(xln(z^3+x^3)) ?
The answer to (\partial)/(\partial x)(xln(z^3+x^3)) is ln(z^3+x^3)+(3x^3)/(z^3+x^3)