What is (\partial)/(\partial x)(xyln(x^2+2y^2)) ?

The answer to (\partial)/(\partial x)(xyln(x^2+2y^2)) is y(ln(x^2+2y^2)+(2x^2)/(x^2+2y^2))

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(\partial)/(\partial x)(xyln(x^2+2y^2))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (x ln (x^{2} + 2 y^{2}))

Apply the Product Rule:

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

f = x, g = ln (x^{2} + 2 y^{2})

= y (\frac{\partial}{\partial x} (x) ln (x^{2} + 2 y^{2}) + \frac{\partial}{\partial x} (ln (x^{2} + 2 y^{2})) x)

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

\frac{\partial}{\partial x} (ln (x^{2} + 2 y^{2})) = \frac{2 x}{x ^{2} + 2 y ^{2}}

= y (1 \cdot ln (x^{2} + 2 y^{2}) + \frac{2 x}{x ^{2} + 2 y ^{2}} x)

Simplify

y (1 \cdot ln (x^{2} + 2 y^{2}) + \frac{2 x}{x ^{2} + 2 y ^{2}} x) : y (ln (x^{2} + 2 y^{2}) + \frac{2 x ^{2}}{x ^{2} + 2 y ^{2}})

= y (ln (x^{2} + 2 y^{2}) + \frac{2 x ^{2}}{x ^{2} + 2 y ^{2}})

x \to + 0 - lim (e^{\frac{7}{x}})

\frac{\partial}{\partial x} (- \frac{1}{( 10 + x \cdot y )})

x (8^{- 2 x})

\frac{3}{2} sin (2 t)

What is (\partial)/(\partial x)(xyln(x^2+2y^2)) ?
The answer to (\partial)/(\partial x)(xyln(x^2+2y^2)) is y(ln(x^2+2y^2)+(2x^2)/(x^2+2y^2))