What is the derivative of ln(x^2y^3) ?

The derivative of ln(x^2y^3) is (2y+3x\frac{dy)/(dx)}{xy}

What is the first derivative of ln(x^2y^3) ?

The first derivative of ln(x^2y^3) is (2y+3x\frac{dy)/(dx)}{xy}

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

\frac{2 y + 3 x \frac{d y}{d x}}{x y}

Solution steps

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

Apply the chain rule:

\frac{1}{x ^{2} y ^{3}} \frac{d}{d x} (x^{2} y^{3})

= \frac{1}{x ^{2} y ^{3}} \frac{d}{d x} (x^{2} y^{3})

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

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

Simplify

\frac{1}{x ^{2} y ^{3}} (2 x y^{3} + 3 y^{2} \frac{d y}{d x} x^{2}) : \frac{2 y + 3 x \frac{d y}{d x}}{x y}

= \frac{2 y + 3 x \frac{d y}{d x}}{x y}

What is the derivative of ln(x^2y^3) ?
The derivative of ln(x^2y^3) is (2y+3x\frac{dy)/(dx)}{xy}
What is the first derivative of ln(x^2y^3) ?
The first derivative of ln(x^2y^3) is (2y+3x\frac{dy)/(dx)}{xy}