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Popular Calculus >

derivative of ln(xy^2)

Solution

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

Solution

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

Popular Examples

derivative of 1/(6x^2)

\frac{d}{d x} (\frac{1}{6 x ^{2}})

integral of 1/((x^2-4x+4)(x^2-5x+6))

\int \frac{1}{( x ^{2} - 4 x + 4 ) ( x ^{2} - 5 x + 6 )} d x

integral of-1sin^3(x)cos^2(x)

\int - 1 sin^{3} (x) cos^{2} (x) d x

area x^2+1,3-x,x=0

a re a x^{2} + 1, 3 - x, x = 0

derivative of ((tan(x)/x)^2)

\frac{d}{d x} ((\frac{tan ( x )}{x})^{2})

Frequently Asked Questions (FAQ)

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