Calculus Examples

$\frac{f (x + h) - f \frac{3}{x + 5}}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $f$ out of $f (x + h) - f \frac{3}{x + 5}$ .

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Step 1.1.1

Factor $f$ out of $- f \frac{3}{x + 5}$ .

$\frac{f (x + h) + f (- 1 \frac{3}{x + 5})}{h}$

Step 1.1.2

Factor $f$ out of $f (x + h) + f (- 1 \frac{3}{x + 5})$ .

$\frac{f (x + h - 1 \frac{3}{x + 5})}{h}$

Step 1.2

Rewrite $- 1 \frac{3}{x + 5}$ as $- \frac{3}{x + 5}$ .

$\frac{f (x + h - \frac{3}{x + 5})}{h}$

Step 1.3

To write $x$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{f (h + \frac{x (x + 5)}{x + 5} - \frac{3}{x + 5})}{h}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{f (h + \frac{x (x + 5) - 3}{x + 5})}{h}$

Step 1.5

To write $h$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{f (\frac{h (x + 5)}{x + 5} + \frac{x (x + 5) - 3}{x + 5})}{h}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{f \frac{h (x + 5) + x (x + 5) - 3}{x + 5}}{h}$

Step 1.7

Rewrite $\frac{h (x + 5) + x (x + 5) - 3}{x + 5}$ in a factored form.

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Step 1.7.1

Apply the distributive property.

$\frac{f \frac{h x + h \cdot 5 + x (x + 5) - 3}{x + 5}}{h}$

Step 1.7.2

Move $5$ to the left of $h$ .

$\frac{f \frac{h x + 5 \cdot h + x (x + 5) - 3}{x + 5}}{h}$

Step 1.7.3

Apply the distributive property.

$\frac{f \frac{h x + 5 h + x \cdot x + x \cdot 5 - 3}{x + 5}}{h}$

Step 1.7.4

Multiply $x$ by $x$ .

$\frac{f \frac{h x + 5 h + x^{2} + x \cdot 5 - 3}{x + 5}}{h}$

Step 1.7.5

Move $5$ to the left of $x$ .

$\frac{f \frac{h x + 5 h + x^{2} + 5 x - 3}{x + 5}}{h}$

Step 2

Combine $f$ and $\frac{h x + 5 h + x^{2} + 5 x - 3}{x + 5}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{f (h x + 5 h + x^{2} + 5 x - 3)}{x + 5}$ by $\frac{1}{h}$ .

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

Step 5

Reorder factors in $\frac{f (h x + 5 h + x^{2} + 5 x - 3)}{(x + 5) h}$ .

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