Solve 1/a-1-3/a+3/a+1-1/a+2

$$\frac { 1 } { a - 1 } - \frac { 3 } { a } + \frac { 3 } { a + 1 } - \frac { 1 } { a + 2 }$$

Evaluate

$$\frac{6}{a\left(a+2\right)\left(a^{2}-1\right)}$$

Short Solution Steps

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

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

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

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

Do the multiplications in a-3\left(a-1\right).

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

Combine like terms in a-3a+3.

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

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

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

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

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

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

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

Combine like terms in -2a^{2}-2a+3a+3+3a^{2}-3a.

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

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

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

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

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

Do the multiplications in \left(a^{2}-2a+3\right)\left(a+2\right)-a\left(a-1\right)\left(a+1\right).

$$\frac{a^{3}+2a^{2}-2a^{2}-4a+3a+6-a^{3}-a^{2}+a^{2}+a}{a\left(a-1\right)\left(a+1\right)\left(a+2\right)}$$

Combine like terms in a^{3}+2a^{2}-2a^{2}-4a+3a+6-a^{3}-a^{2}+a^{2}+a.

$$\frac{6}{a\left(a-1\right)\left(a+1\right)\left(a+2\right)}$$

Expand a\left(a-1\right)\left(a+1\right)\left(a+2\right).

$$\frac{6}{a^{4}+2a^{3}-a^{2}-2a}$$

Differentiate w.r.t. a

$$\frac{12\left(2a+1\right)\left(1-a-a^{2}\right)}{\left(a\left(a+2\right)\left(a^{2}-1\right)\right)^{2}}$$