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

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

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

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

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

Solution steps

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

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

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

\frac{\partial}{\partial x} (ln (z^{5} + x^{7})) = \frac{7 x ^{6}}{z ^{5} + x ^{7}}

= 1 \cdot ln (z^{5} + x^{7}) + \frac{7 x ^{6}}{z ^{5} + x ^{7}} x

Simplify

1 \cdot ln (z^{5} + x^{7}) + \frac{7 x ^{6}}{z ^{5} + x ^{7}} x : ln (z^{5} + x^{7}) + \frac{7 x ^{7}}{z ^{5} + x ^{7}}

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

x y y^{^{'}} + 4 x^{2} + y^{2} = 0

y^{^{''}} + 6 y^{^{'}} + 10 = 0

\frac{d y}{d x} = \frac{3 x + 2 y}{3 x + 2 y + 2}

\int 6 x^{2} + 8 x^{3} + C d x

2 x d z - 2 z d x = x^{2} + 4 z^{2} d x

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