What is (\partial)/(\partial r)(rln(r^2+s^2)) ?

The answer to (\partial)/(\partial r)(rln(r^2+s^2)) is ln(r^2+s^2)+(2r^2)/(r^2+s^2)

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(\partial)/(\partial r)(rln(r^2+s^2))

\frac{\partial}{\partial r} (r ln (r^{2} + s^{2}))

ln (r^{2} + s^{2}) + \frac{2 r ^{2}}{r ^{2} + s ^{2}}

Solution steps

\frac{\partial}{\partial r} (r ln (r^{2} + s^{2}))

Treat

s

as a constant

Apply the Product Rule:

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

f = r, g = ln (r^{2} + s^{2})

= \frac{\partial}{\partial r} (r) ln (r^{2} + s^{2}) + \frac{\partial}{\partial r} (ln (r^{2} + s^{2})) r

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

\frac{\partial}{\partial r} (ln (r^{2} + s^{2})) = \frac{2 r}{r ^{2} + s ^{2}}

= 1 \cdot ln (r^{2} + s^{2}) + \frac{2 r}{r ^{2} + s ^{2}} r

Simplify

1 \cdot ln (r^{2} + s^{2}) + \frac{2 r}{r ^{2} + s ^{2}} r : ln (r^{2} + s^{2}) + \frac{2 r ^{2}}{r ^{2} + s ^{2}}

= ln (r^{2} + s^{2}) + \frac{2 r ^{2}}{r ^{2} + s ^{2}}

\int (3 x^{2} - 8 x) d x

\int \frac{2 + x}{x} d x

x \to 1 - lim (x + 1)

\int ln (5 t) d t

y = x^{2} - 3 x, (3, 0)

What is (\partial)/(\partial r)(rln(r^2+s^2)) ?
The answer to (\partial)/(\partial r)(rln(r^2+s^2)) is ln(r^2+s^2)+(2r^2)/(r^2+s^2)