Solve 2d/frac{d{r}+d/s}=

$$\frac { 2 d } { \frac { d } { r } + \frac { d } { s } } =$$

Evaluate

$$\frac{2rs}{r+s}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r and s is rs. Multiply \frac{d}{r} times \frac{s}{s}. Multiply \frac{d}{s} times \frac{r}{r}.

$$\frac{2d}{\frac{ds}{rs}+\frac{dr}{rs}}$$

Since \frac{ds}{rs} and \frac{dr}{rs} have the same denominator, add them by adding their numerators.

$$\frac{2d}{\frac{ds+dr}{rs}}$$

Divide 2d by \frac{ds+dr}{rs} by multiplying 2d by the reciprocal of \frac{ds+dr}{rs}.

$$\frac{2drs}{ds+dr}$$

Factor the expressions that are not already factored.

$$\frac{2drs}{d\left(r+s\right)}$$

Cancel out d in both numerator and denominator.

$$\frac{2rs}{r+s}$$

Differentiate w.r.t. r

$$2\times \left(\frac{s}{r+s}\right)^{2}$$