What is (\partial)/(\partial x)(ln(ax+by)) ?

The answer to (\partial)/(\partial x)(ln(ax+by)) is a/(ax+by)

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(\partial)/(\partial x)(ln(ax+by))

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

\frac{a}{a x + b y}

Solution steps

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

Treat

a, b, y

as constants

Apply the chain rule:

\frac{1}{a x + b y} \frac{\partial}{\partial x} (a x + b y)

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

\frac{\partial}{\partial x} (a x + b y) = a

= \frac{1}{a x + b y} a

Simplify

\frac{1}{a x + b y} a : \frac{a}{a x + b y}

= \frac{a}{a x + b y}

n = 1 \sum \infty (1)^{n}

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

\int \frac{4 x ^{2} + 7 x + 4}{( x ^{2} + 1 ) ^{2}} d x

\frac{d}{d x} (\frac{x ^{3} - 5 x ^{2}}{x ^{8} - 4})

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

What is (\partial)/(\partial x)(ln(ax+by)) ?
The answer to (\partial)/(\partial x)(ln(ax+by)) is a/(ax+by)