Precalculus Examples

$f (x) = \tan (2 x)$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \tan (2 (x + h))$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = \tan (2 x + 2 h)$

Step 2.1.2.2

The final answer is $\tan (2 x + 2 h)$ .

$\tan (2 x + 2 h)$

Step 2.2

Find the components of the definition.

$f (x + h) = \tan (2 x + 2 h)$

$f (x) = \tan (2 x)$

$f (x + h) = \tan (2 x + 2 h)$

$f (x) = \tan (2 x)$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\tan (2 x + 2 h) - (\tan (2 x))}{h}$

Step 4

Simplify.

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Step 4.1

Use a sum or difference formula on the numerator.

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Step 4.1.1

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

$\frac{(\frac{\tan (2 x) + \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}) - (\tan (2 x))}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

To write $- \tan (2 x)$ as a fraction with a common denominator, multiply by $\frac{1 - \tan (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}$ .

$\frac{\frac{\tan (2 x) + \tan (2 h)}{1 - \tan (2 x) \tan (2 h)} - \tan (2 x) \cdot \frac{1 - \tan (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $- \tan (2 x)$ and $\frac{1 - \tan (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}$ .

$\frac{\frac{\tan (2 x) + \tan (2 h)}{1 - \tan (2 x) \tan (2 h)} + \frac{- \tan (2 x) (1 - \tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) (1 - \tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Apply the distributive property.

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) \cdot 1 - \tan (2 x) (- \tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.2

Multiply $- 1$ by $1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) - \tan (2 x) (- \tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3

Multiply $- \tan (2 x) (- \tan (2 x) \tan (2 h))$ .

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Step 4.1.2.3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + 1 \tan (2 x) (\tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3.2

Multiply $\tan (2 x)$ by $1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + \tan (2 x) (\tan (2 x) \tan (2 h))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3.3

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + \tan (2 x) \tan (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3.4

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + \tan (2 x) \tan (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3.5

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

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + {\tan (2 x)}^{1 + 1} \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.3.6

Add $1$ and $1$ .

$\frac{\frac{\tan (2 x) + \tan (2 h) - \tan (2 x) + \tan^{2} (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.4

Subtract $\tan (2 x)$ from $\tan (2 x)$ .

$\frac{\frac{0 + \tan (2 h) + \tan^{2} (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.5

Add $0$ and $\tan (2 h)$ .

$\frac{\frac{\tan (2 h) + \tan^{2} (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.6

Factor $\tan (2 h)$ out of $\tan (2 h) + \tan^{2} (2 x) \tan (2 h)$ .

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Step 4.1.2.3.6.1

Multiply by $1$ .

$\frac{\frac{\tan (2 h) \cdot 1 + \tan^{2} (2 x) \tan (2 h)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.6.2

Factor $\tan (2 h)$ out of $\tan^{2} (2 x) \tan (2 h)$ .

$\frac{\frac{\tan (2 h) \cdot 1 + \tan (2 h) \tan^{2} (2 x)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.6.3

Factor $\tan (2 h)$ out of $\tan (2 h) \cdot 1 + \tan (2 h) \tan^{2} (2 x)$ .

$\frac{\frac{\tan (2 h) (1 + \tan^{2} (2 x))}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.7

Rearrange terms.

$\frac{\frac{\tan (2 h) (\tan^{2} (2 x) + 1)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.1.2.3.8

Apply pythagorean identity.

$\frac{\frac{\tan (2 h) \sec^{2} (2 x)}{1 - \tan (2 x) \tan (2 h)}}{h}$

Step 4.2

Simplify.

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Step 4.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\tan (2 h) \sec^{2} (2 x)}{1 - \tan (2 x) \tan (2 h)} \cdot \frac{1}{h}$

Step 4.2.2

Multiply $\frac{\tan (2 h) \sec^{2} (2 x)}{1 - \tan (2 x) \tan (2 h)}$ by $\frac{1}{h}$ .

$\frac{\tan (2 h) \sec^{2} (2 x)}{(1 - \tan (2 x) \tan (2 h)) h}$

Step 4.2.3

Reorder factors in $\frac{\tan (2 h) \sec^{2} (2 x)}{(1 - \tan (2 x) \tan (2 h)) h}$ .

$\frac{\tan (2 h) \sec^{2} (2 x)}{h (1 - \tan (2 x) \tan (2 h))}$

Step 5