What is the derivative of ln(x*y) ?

The derivative of ln(x*y) is (y+x\frac{dy)/(dx)}{xy}

What is the first derivative of ln(x*y) ?

The first derivative of ln(x*y) is (y+x\frac{dy)/(dx)}{xy}

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\frac{d}{d x} (ln (x \cdot y))

\frac{y + x \frac{d y}{d x}}{x y}

Solution steps

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

Apply the chain rule:

\frac{1}{x y} \frac{d}{d x} (x y)

= \frac{1}{x y} \frac{d}{d x} (x y)

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

= \frac{1}{x y} (y + x \frac{d y}{d x})

Simplify

\frac{1}{x y} (y + x \frac{d y}{d x}) : \frac{y + x \frac{d y}{d x}}{x y}

= \frac{y + x \frac{d y}{d x}}{x y}

\int x^{4} e^{x^{5}} d x

5 x sin (x^{2})

\int \frac{e ^{x}}{x} d x

\int_{0}^{3} 3^{x} d x

\frac{3}{x + 1}

What is the derivative of ln(x*y) ?
The derivative of ln(x*y) is (y+x\frac{dy)/(dx)}{xy}
What is the first derivative of ln(x*y) ?
The first derivative of ln(x*y) is (y+x\frac{dy)/(dx)}{xy}