Solve 2/n+1+3/k+1=

$$\frac { 2 } { n + 1 } + \frac { 3 } { k + 1 } =$$

Evaluate

$$\frac{3n+2k+5}{\left(k+1\right)\left(n+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n+1 and k+1 is \left(k+1\right)\left(n+1\right). Multiply \frac{2}{n+1} times \frac{k+1}{k+1}. Multiply \frac{3}{k+1} times \frac{n+1}{n+1}.

$$\frac{2\left(k+1\right)}{\left(k+1\right)\left(n+1\right)}+\frac{3\left(n+1\right)}{\left(k+1\right)\left(n+1\right)}$$

Since \frac{2\left(k+1\right)}{\left(k+1\right)\left(n+1\right)} and \frac{3\left(n+1\right)}{\left(k+1\right)\left(n+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(k+1\right)+3\left(n+1\right)}{\left(k+1\right)\left(n+1\right)}$$

Do the multiplications in 2\left(k+1\right)+3\left(n+1\right).

$$\frac{2k+2+3n+3}{\left(k+1\right)\left(n+1\right)}$$

Combine like terms in 2k+2+3n+3.

$$\frac{2k+5+3n}{\left(k+1\right)\left(n+1\right)}$$

Expand \left(k+1\right)\left(n+1\right).

$$\frac{2k+5+3n}{kn+n+k+1}$$

Differentiate w.r.t. k

$$-\frac{3}{\left(k+1\right)^{2}}$$