Precalculus Examples

$y = 3 \sin (2 x)$

Step 1

Write $y = 3 \sin (2 x)$ as a function.

$f (x) = 3 \sin (2 x)$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = 3 \sin (2 (x + h))$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = 3 \sin (2 x + 2 h)$

Step 3.1.2.2

The final answer is $3 \sin (2 x + 2 h)$ .

$3 \sin (2 x + 2 h)$

Step 3.2

Find the components of the definition.

$f (x + h) = 3 \sin (2 x + 2 h)$

$f (x) = 3 \sin (2 x)$

$f (x + h) = 3 \sin (2 x + 2 h)$

$f (x) = 3 \sin (2 x)$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{3 \sin (2 x + 2 h) - (3 \sin (2 x))}{h}$

Step 5

Simplify.

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Step 5.1

Use a sum or difference formula on the numerator.

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Step 5.1.1

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

$\frac{(3 (3 (\sin (2 x) \cos (2 h) + \cos (2 x) \sin (2 h)))) - (3 \sin (2 x))}{h}$

Step 5.1.2

Simplify by multiplying through.

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Step 5.1.2.1

Apply the distributive property.

$\frac{(3 (3 (\sin (2 x) \cos (2 h)) + 3 (\cos (2 x) \sin (2 h)))) - (3 \sin (2 x))}{h}$

Step 5.1.2.2

Apply the distributive property.

$\frac{(3 (3 \sin (2 x) \cos (2 h)) + 3 (3 \cos (2 x) \sin (2 h))) - (3 \sin (2 x))}{h}$

Step 5.1.2.3

Multiply.

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Step 5.1.2.3.1

Multiply $3$ by $3$ .

$\frac{(9 (\sin (2 x) \cos (2 h)) + 3 (3 \cos (2 x) \sin (2 h))) - (3 \sin (2 x))}{h}$

Step 5.1.2.3.2

Multiply $3$ by $3$ .

$\frac{(9 (\sin (2 x) \cos (2 h)) + 9 (\cos (2 x) \sin (2 h))) - (3 \sin (2 x))}{h}$

Step 5.1.3

Remove parentheses.

$\frac{(9 \sin (2 x) \cos (2 h) + 9 \cos (2 x) \sin (2 h)) - (3 \sin (2 x))}{h}$

Step 5.1.4

Multiply $3$ by $- 1$ .

$\frac{9 \sin (2 x) \cos (2 h) + 9 \cos (2 x) \sin (2 h) - 3 \sin (2 x)}{h}$

Step 5.2

Factor $3$ out of $9 \sin (2 x) \cos (2 h) + 9 \cos (2 x) \sin (2 h) - 3 \sin (2 x)$ .

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Step 5.2.1

Factor $3$ out of $9 \sin (2 x) \cos (2 h)$ .

$\frac{3 (3 \sin (2 x) \cos (2 h)) + 9 \cos (2 x) \sin (2 h) - 3 \sin (2 x)}{h}$

Step 5.2.2

Factor $3$ out of $9 \cos (2 x) \sin (2 h)$ .

$\frac{3 (3 \sin (2 x) \cos (2 h)) + 3 (3 \cos (2 x) \sin (2 h)) - 3 \sin (2 x)}{h}$

Step 5.2.3

Factor $3$ out of $- 3 \sin (2 x)$ .

$\frac{3 (3 \sin (2 x) \cos (2 h)) + 3 (3 \cos (2 x) \sin (2 h)) + 3 (- \sin (2 x))}{h}$

Step 5.2.4

Factor $3$ out of $3 (3 \sin (2 x) \cos (2 h)) + 3 (3 \cos (2 x) \sin (2 h))$ .

$\frac{3 (3 \sin (2 x) \cos (2 h) + 3 \cos (2 x) \sin (2 h)) + 3 (- \sin (2 x))}{h}$

Step 5.2.5

Factor $3$ out of $3 (3 \sin (2 x) \cos (2 h) + 3 \cos (2 x) \sin (2 h)) + 3 (- \sin (2 x))$ .

$\frac{3 (3 \sin (2 x) \cos (2 h) + 3 \cos (2 x) \sin (2 h) - \sin (2 x))}{h}$

Step 6