What is (\partial)/(\partial x)(xln(z^5+x^4)) ?

The answer to (\partial)/(\partial x)(xln(z^5+x^4)) is ln(z^5+x^4)+(4x^4)/(z^5+x^4)

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(\partial)/(\partial x)(xln(z^5+x^4))

\frac{\partial}{\partial x} (x ln (z^{5} + x^{4}))

ln (z^{5} + x^{4}) + \frac{4 x ^{4}}{z ^{5} + x ^{4}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{5} + x^{4}))

Treat

z

as a constant

Apply the Product Rule:

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

f = x, g = ln (z^{5} + x^{4})

= \frac{\partial}{\partial x} (x) ln (z^{5} + x^{4}) + \frac{\partial}{\partial x} (ln (z^{5} + x^{4})) x

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

\frac{\partial}{\partial x} (ln (z^{5} + x^{4})) = \frac{4 x ^{3}}{z ^{5} + x ^{4}}

= 1 \cdot ln (z^{5} + x^{4}) + \frac{4 x ^{3}}{z ^{5} + x ^{4}} x

Simplify

1 \cdot ln (z^{5} + x^{4}) + \frac{4 x ^{3}}{z ^{5} + x ^{4}} x : ln (z^{5} + x^{4}) + \frac{4 x ^{4}}{z ^{5} + x ^{4}}

= ln (z^{5} + x^{4}) + \frac{4 x ^{4}}{z ^{5} + x ^{4}}

\frac{\partial}{\partial x} (7 x y^{2} + 1)

\frac{d}{d x} (4 ln (2 x))

x \to 0 lim ((1 + 5 x)^{\frac{3}{x}})

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

What is (\partial)/(\partial x)(xln(z^5+x^4)) ?
The answer to (\partial)/(\partial x)(xln(z^5+x^4)) is ln(z^5+x^4)+(4x^4)/(z^5+x^4)