What is (\partial)/(\partial x)(xln(z^7+x^5)) ?

The answer to (\partial)/(\partial x)(xln(z^7+x^5)) is ln(z^7+x^5)+(5x^5)/(z^7+x^5)

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(\partial)/(\partial x)(xln(z^7+x^5))

\frac{\partial}{\partial x} (x ln (z^{7} + x^{5}))

ln (z^{7} + x^{5}) + \frac{5 x ^{5}}{z ^{7} + x ^{5}}

Solution steps

\frac{\partial}{\partial x} (x ln (z^{7} + x^{5}))

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^{5})

= \frac{\partial}{\partial x} (x) ln (z^{7} + x^{5}) + \frac{\partial}{\partial x} (ln (z^{7} + x^{5})) x

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

\frac{\partial}{\partial x} (ln (z^{7} + x^{5})) = \frac{5 x ^{4}}{z ^{7} + x ^{5}}

= 1 \cdot ln (z^{7} + x^{5}) + \frac{5 x ^{4}}{z ^{7} + x ^{5}} x

Simplify

1 \cdot ln (z^{7} + x^{5}) + \frac{5 x ^{4}}{z ^{7} + x ^{5}} x : ln (z^{7} + x^{5}) + \frac{5 x ^{5}}{z ^{7} + x ^{5}}

= ln (z^{7} + x^{5}) + \frac{5 x ^{5}}{z ^{7} + x ^{5}}

f (x) = (x^{2} + 4 x + 3)^{2}

n = 1 \sum \infty \frac{n}{2 ^{n - 1}}

y^{^{'}} - \frac{1}{x} y = 1

y = \frac{1}{4 + 3 x}, (1, \frac{1}{7})

f^{^{'}} (x) = 2 - 4

What is (\partial)/(\partial x)(xln(z^7+x^5)) ?
The answer to (\partial)/(\partial x)(xln(z^7+x^5)) is ln(z^7+x^5)+(5x^5)/(z^7+x^5)