Solve 3/x-2/x+1+1/x^2=

$$\frac { 3 } { x } - \frac { 2 } { x + 1 } + \frac { 1 } { x ^ { 2 } } =$$

Evaluate

$$\frac{x^{2}+4x+1}{\left(x+1\right)x^{2}}$$

Short Solution Steps

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

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

Since \frac{3\left(x+1\right)}{x\left(x+1\right)} and \frac{2x}{x\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x+1\right)-2x.

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

Combine like terms in 3x+3-2x.

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

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

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

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

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

Do the multiplications in \left(x+3\right)x+x+1.

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

Combine like terms in x^{2}+3x+x+1.

$$\frac{x^{2}+4x+1}{\left(x+1\right)x^{2}}$$

Expand \left(x+1\right)x^{2}.

$$\frac{x^{2}+4x+1}{x^{3}+x^{2}}$$

Differentiate w.r.t. x

$$-\frac{x^{3}+8x^{2}+7x+2}{\left(x+1\right)^{2}x^{3}}$$