What is (\partial)/(\partial x)(x^2ln(xy)) ?

The answer to (\partial)/(\partial x)(x^2ln(xy)) is 2xln(xy)+x

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(\partial)/(\partial x)(x^2ln(xy))

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

2 x ln (x y) + x

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

\frac{1}{x} x^{2} = x

= 2 x ln (x y) + x

\int_{1}^{x} e^{x^{2}} d x

\frac{d}{d x} (12 x^{2} - 12 x)

\frac{d}{d j} (4 i + 5 j)

\frac{e ^{x}}{3 x ^{2}}

\frac{d}{d x} ((4 x^{3} + 3)^{2} (5 x^{2} + 3))

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