What is (\partial)/(\partial x)(xln(z^7+x^3)) ?

The answer to (\partial)/(\partial x)(xln(z^7+x^3)) is ln(z^7+x^3)+(3x^3)/(z^7+x^3)

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(\partial)/(\partial x)(xln(z^7+x^3))

\frac{\partial}{\partial x} (x ln (z^{7} + x^{3}))

ln (z^{7} + x^{3}) + \frac{3 x ^{3}}{z ^{7} + x ^{3}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{7} + 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^{7} + x^{3})

= \frac{\partial}{\partial x} (x) ln (z^{7} + x^{3}) + \frac{\partial}{\partial x} (ln (z^{7} + x^{3})) x

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

\frac{\partial}{\partial x} (ln (z^{7} + x^{3})) = \frac{3 x ^{2}}{z ^{7} + x ^{3}}

= 1 \cdot ln (z^{7} + x^{3}) + \frac{3 x ^{2}}{z ^{7} + x ^{3}} x

Simplify

1 \cdot ln (z^{7} + x^{3}) + \frac{3 x ^{2}}{z ^{7} + x ^{3}} x : ln (z^{7} + x^{3}) + \frac{3 x ^{3}}{z ^{7} + x ^{3}}

= ln (z^{7} + x^{3}) + \frac{3 x ^{3}}{z ^{7} + x ^{3}}

y = \frac{1}{2 x - 2}

x \to 7 lim (\frac{2 x + 1}{x - 7})

\int 2 cos (5 x) d x

\frac{d y}{d x} = y (4 x^{2} + 1)

y = \frac{x}{3}

What is (\partial)/(\partial x)(xln(z^7+x^3)) ?
The answer to (\partial)/(\partial x)(xln(z^7+x^3)) is ln(z^7+x^3)+(3x^3)/(z^7+x^3)