What is (\partial)/(\partial x)(xln(z^4+x^7)) ?

The answer to (\partial)/(\partial x)(xln(z^4+x^7)) is ln(z^4+x^7)+(7x^7)/(z^4+x^7)

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(\partial)/(\partial x)(xln(z^4+x^7))

\frac{\partial}{\partial x} (x ln (z^{4} + x^{7}))

ln (z^{4} + x^{7}) + \frac{7 x ^{7}}{z ^{4} + x ^{7}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{4} + x^{7}))

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^{7})

= \frac{\partial}{\partial x} (x) ln (z^{4} + x^{7}) + \frac{\partial}{\partial x} (ln (z^{4} + x^{7})) x

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

\frac{\partial}{\partial x} (ln (z^{4} + x^{7})) = \frac{7 x ^{6}}{z ^{4} + x ^{7}}

= 1 \cdot ln (z^{4} + x^{7}) + \frac{7 x ^{6}}{z ^{4} + x ^{7}} x

Simplify

1 \cdot ln (z^{4} + x^{7}) + \frac{7 x ^{6}}{z ^{4} + x ^{7}} x : ln (z^{4} + x^{7}) + \frac{7 x ^{7}}{z ^{4} + x ^{7}}

= ln (z^{4} + x^{7}) + \frac{7 x ^{7}}{z ^{4} + x ^{7}}

What is (\partial)/(\partial x)(xln(z^4+x^7)) ?
The answer to (\partial)/(\partial x)(xln(z^4+x^7)) is ln(z^4+x^7)+(7x^7)/(z^4+x^7)