Calculus Examples

$\frac{3}{x - 1} - \frac{5}{{(x - 1)}^{2}}$

Step 1

To write $\frac{3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{3}{x - 1} \cdot \frac{x - 1}{x - 1} - \frac{5}{{(x - 1)}^{2}}$

Step 2

Write each expression with a common denominator of ${(x - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{3}{x - 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{3 (x - 1)}{(x - 1) (x - 1)} - \frac{5}{{(x - 1)}^{2}}$

Step 2.2

Raise $x - 1$ to the power of $1$ .

$\frac{3 (x - 1)}{{(x - 1)}^{1} (x - 1)} - \frac{5}{{(x - 1)}^{2}}$

Step 2.3

Raise $x - 1$ to the power of $1$ .

$\frac{3 (x - 1)}{{(x - 1)}^{1} {(x - 1)}^{1}} - \frac{5}{{(x - 1)}^{2}}$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (x - 1)}{{(x - 1)}^{1 + 1}} - \frac{5}{{(x - 1)}^{2}}$

Step 2.5

Add $1$ and $1$ .

$\frac{3 (x - 1)}{{(x - 1)}^{2}} - \frac{5}{{(x - 1)}^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 (x - 1) - 5}{{(x - 1)}^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 1 - 5}{{(x - 1)}^{2}}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{3 x - 3 - 5}{{(x - 1)}^{2}}$

Step 4.3

Subtract $5$ from $- 3$ .

$\frac{3 x - 8}{{(x - 1)}^{2}}$