Solve 5/a-1-5/a^2-a

$$\frac { 5 } { a - 1 } - \frac { 5 } { a ^ { 2 } - a }$$

Evaluate

$$\frac{5}{a}$$

Short Solution Steps

Factor a^{2}-a.

$$\frac{5}{a-1}-\frac{5}{a\left(a-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and a\left(a-1\right) is a\left(a-1\right). Multiply \frac{5}{a-1} times \frac{a}{a}.

$$\frac{5a}{a\left(a-1\right)}-\frac{5}{a\left(a-1\right)}$$

Since \frac{5a}{a\left(a-1\right)} and \frac{5}{a\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5a-5}{a\left(a-1\right)}$$

Factor the expressions that are not already factored in \frac{5a-5}{a\left(a-1\right)}.

$$\frac{5\left(a-1\right)}{a\left(a-1\right)}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{5}{a}$$

Differentiate w.r.t. a

$$-\frac{5}{a^{2}}$$

Steps Using Definition of a Derivative

Factor a^{2}-a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5}{a-1}-\frac{5}{a\left(a-1\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and a\left(a-1\right) is a\left(a-1\right). Multiply \frac{5}{a-1} times \frac{a}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5a}{a\left(a-1\right)}-\frac{5}{a\left(a-1\right)})$$

Since \frac{5a}{a\left(a-1\right)} and \frac{5}{a\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5a-5}{a\left(a-1\right)})$$

Factor the expressions that are not already factored in \frac{5a-5}{a\left(a-1\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5\left(a-1\right)}{a\left(a-1\right)})$$

Cancel out a-1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5}{a})$$

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

$$-5a^{-1-1}$$

Subtract 1 from -1.

$$-5a^{-2}$$