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Popular >

derivative of ln(sqrt(xy))

Solution

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

Solution

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

= \frac{1}{x y} \cdot \frac{y + x \frac{d y}{d x}}{2 x y}

Simplify

\frac{1}{x y} \cdot \frac{y + x \frac{d y}{d x}}{2 x y} : \frac{y + x \frac{d y}{d x}}{2 x y}

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

Popular Examples

d/(dw)((ln(w))^{(2w)})

\frac{d}{d w} ((ln (w))^{(2 w)})

integral from 0 to pi of 4*sin(x)

\int_{0}^{π} 4 \cdot sin (x) d x

integral of 1/((1+2x)^2)

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

integral of 1+e^{x^2}

\int 1 + e^{x^{2}} d x

(\partial)/(\partial y)(e^{x^2+y^2}*2y)

\frac{\partial}{\partial y} (e^{x^{2} + y^{2}} \cdot 2 y)

Frequently Asked Questions (FAQ)

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