What is (\partial)/(\partial x)(xln((x^2)/(y^3))) ?

The answer to (\partial)/(\partial x)(xln((x^2)/(y^3))) is 2ln(x)-3ln(y)+2

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(\partial)/(\partial x)(xln((x^2)/(y^3)))

\frac{\partial}{\partial x} (x ln (\frac{x ^{2}}{y ^{3}}))

2 ln (x) - 3 ln (y) + 2

Solution steps

\frac{\partial}{\partial x} (x ln (\frac{x ^{2}}{y ^{3}}))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (\frac{x ^{2}}{y ^{3}})

= \frac{\partial}{\partial x} (x) ln (\frac{x ^{2}}{y ^{3}}) + \frac{\partial}{\partial x} (ln (\frac{x ^{2}}{y ^{3}})) x

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

\frac{\partial}{\partial x} (ln (\frac{x ^{2}}{y ^{3}})) = \frac{2}{x}

= 1 \cdot ln (\frac{x ^{2}}{y ^{3}}) + \frac{2}{x} x

Simplify

1 \cdot ln (\frac{x ^{2}}{y ^{3}}) + \frac{2}{x} x : 2 ln (x) - 3 ln (y) + 2

= 2 ln (x) - 3 ln (y) + 2

\int_{- 2}^{3} \frac{26}{x ^{4}} d x

\frac{d}{d x} (\frac{1}{1 + x ^{4}})

\frac{d}{d x} (4 sec (x) - 8 cos (x))

y = x^{3}, y = x, x \geq 0

\int_{0}^{1} x^{9} e^{x^{5}} d x

What is (\partial)/(\partial x)(xln((x^2)/(y^3))) ?
The answer to (\partial)/(\partial x)(xln((x^2)/(y^3))) is 2ln(x)-3ln(y)+2