Precalculus Examples

$f (2) = 3^{6} + 25$

Step 1

Write $f (2) - 754 = 0$ as a function.

$f (x) = f (2) - 754$

Step 2

Move all the expressions to the left side of the equation.

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Step 2.1

Subtract $3^{6}$ from both sides of the equation.

$f (2) - 3^{6} = 25$

Step 2.2

Subtract $25$ from both sides of the equation.

$f (2) - 3^{6} - 25 = 0$

Step 3

Simplify $f (2) - 3^{6} - 25$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $3$ to the power of $6$ .

$f (2) - 1 \cdot 729 - 25 = 0$

Step 3.1.2

Multiply $- 1$ by $729$ .

$f (2) - 729 - 25 = 0$

Step 3.2

Subtract $25$ from $- 729$ .

$f (2) - 754 = 0$

Step 4

Consider the difference quotient formula.

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

Step 5

Find the components of the definition.

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Step 5.1

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

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Step 5.1.1

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

$f (x + h) = f (2) - 754$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Move $2$ to the left of $f$ .

$f (x + h) = 2 f - 754$

Step 5.1.2.2

The final answer is $2 f - 754$ .

$2 f - 754$

Step 5.2

Find the components of the definition.

$f (x + h) = 2 f - 754$

$f (x) = 2 f - 754$

$f (x + h) = 2 f - 754$

$f (x) = 2 f - 754$

Step 6

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 f - 754 - (2 f - 754)}{h}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$\frac{2 f - 754 - (2 f) - - 754}{h}$

Step 7.1.2

Multiply $2$ by $- 1$ .

$\frac{2 f - 754 - 2 f - - 754}{h}$

Step 7.1.3

Multiply $- 1$ by $- 754$ .

$\frac{2 f - 754 - 2 f + 754}{h}$

Step 7.1.4

Subtract $2 f$ from $2 f$ .

$\frac{0 - 754 + 754}{h}$

Step 7.1.5

Subtract $754$ from $0$ .

$\frac{- 754 + 754}{h}$

Step 7.1.6

Add $- 754$ and $754$ .

$\frac{0}{h}$

Step 7.2

Divide $0$ by $h$ .

$0$

Step 8