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

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

Evaluate

$$\frac{1}{a}$$

Short Solution Steps

Factor a^{2}-2a.

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

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

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

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

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

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

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

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

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

Factor a^{2}-3a+2.

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

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

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

Since \frac{a^{2}-4a+2}{a\left(a-2\right)\left(a-1\right)} and \frac{a}{a\left(a-2\right)\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

Factor the expressions that are not already factored in \frac{a^{2}-3a+2}{a\left(a-2\right)\left(a-1\right)}.

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

Cancel out \left(a-2\right)\left(a-1\right) in both numerator and denominator.

$$\frac{1}{a}$$

Differentiate w.r.t. a

$$-\frac{1}{a^{2}}$$

Steps Using Definition of a Derivative

Factor a^{2}-2a.

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

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

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

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

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

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

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

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

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

Factor a^{2}-3a+2.

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

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

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

Since \frac{a^{2}-4a+2}{a\left(a-2\right)\left(a-1\right)} and \frac{a}{a\left(a-2\right)\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

Factor the expressions that are not already factored in \frac{a^{2}-3a+2}{a\left(a-2\right)\left(a-1\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(a-2\right)\left(a-1\right)}{a\left(a-2\right)\left(a-1\right)})$$

Cancel out \left(a-2\right)\left(a-1\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a})$$

The derivative of ax^{n} is nax^{n-1}.

$$-a^{-1-1}$$

Subtract 1 from -1.

$$-a^{-2}$$