What is (\partial)/(\partial x)(xln(z^4+x^3)) ?

The answer to (\partial)/(\partial x)(xln(z^4+x^3)) is ln(z^4+x^3)+(3x^3)/(z^4+x^3)

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(\partial)/(\partial x)(xln(z^4+x^3))

\frac{\partial}{\partial x} (x ln (z^{4} + x^{3}))

ln (z^{4} + x^{3}) + \frac{3 x ^{3}}{z ^{4} + x ^{3}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{4} + 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^{4} + x^{3})

= \frac{\partial}{\partial x} (x) ln (z^{4} + x^{3}) + \frac{\partial}{\partial x} (ln (z^{4} + x^{3})) x

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

\frac{\partial}{\partial x} (ln (z^{4} + x^{3})) = \frac{3 x ^{2}}{z ^{4} + x ^{3}}

= 1 \cdot ln (z^{4} + x^{3}) + \frac{3 x ^{2}}{z ^{4} + x ^{3}} x

Simplify

1 \cdot ln (z^{4} + x^{3}) + \frac{3 x ^{2}}{z ^{4} + x ^{3}} x : ln (z^{4} + x^{3}) + \frac{3 x ^{3}}{z ^{4} + x ^{3}}

= ln (z^{4} + x^{3}) + \frac{3 x ^{3}}{z ^{4} + x ^{3}}

f (x) = (\frac{x + 3}{x - 3})^{2}, at x = 4

\frac{d}{d x} (ln (\frac{( 1 - x )}{1 + x}))

\int_{1}^{9} π (x - 1)^{2} d x

s \to 0 lim (\frac{3}{s \cdot ( s + 2 )})

\frac{x ^{3}}{9}

What is (\partial)/(\partial x)(xln(z^4+x^3)) ?
The answer to (\partial)/(\partial x)(xln(z^4+x^3)) is ln(z^4+x^3)+(3x^3)/(z^4+x^3)