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

The answer to (\partial)/(\partial x)(xln(x/(y^2))) is ln(x)-2ln(y)+1

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(\partial)/(\partial x)(xln(x/(y^2)))

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

ln (x) - 2 ln (y) + 1

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= ln (x) - 2 ln (y) + 1

n \to \infty lim (ln (2))

\frac{d y}{d x} + 4 x y = 8 x^{3}

\frac{d}{d x} (\frac{cos ( x )}{5})

x \to \infty lim (\frac{2 x}{1})

\int \frac{2 x + 3}{x ( x ^{2} - 4 ) ^{2}} d x

What is (\partial)/(\partial x)(xln(x/(y^2))) ?
The answer to (\partial)/(\partial x)(xln(x/(y^2))) is ln(x)-2ln(y)+1