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

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

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(\partial)/(\partial x)(ln(x^y)+ln(y^x))

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

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

Solution steps

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

Simplify

ln (x^{y}) + ln (y^{x}) : ln (x^{y} y^{x})

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

Treat

y

as a constant

Apply the chain rule:

\frac{1}{x ^{y} y ^{x}} \frac{\partial}{\partial x} (x^{y} y^{x})

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

\frac{\partial}{\partial x} (x^{y} y^{x}) = y^{1 + x} x^{y - 1} + y^{x} x^{y} ln (y)

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

Simplify

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

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

\frac{d}{d x} (cos (3 x) 3)

f (x) = x^{2} - x, at x = - 1

\int (a^{2} + x^{2})^{\frac{1}{2}} d x

y^{^{''}} - 8 y^{^{'}} + 9 y = x e^{x}

sin (2 (t - \frac{π}{2}))

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