What is (\partial)/(\partial x)(xln(y+x)) ?

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

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(\partial)/(\partial x)(xln(y+x))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (y + x)

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

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

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

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

Simplify

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

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

y^{^{'}} = e (- 2 t) - y

f (x) = (x^{3} + 5 x^{2} - 3) x

(cos (3 x))^{^{'}}

\frac{d}{d x} (e^{x} - 1 - x)

\frac{d y}{d x} = e^{- x^{2}}

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