Calculus Examples

$f (x) = - 6 x$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

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.

$f (x + h) = - 6 (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = - 6 x - 6 h$

Step 2.1.2.2

The final answer is $- 6 x - 6 h$ .

$- 6 x - 6 h$

Step 2.2

Find the components of the definition.

$f (x + h) = - 6 x - 6 h$

$f (x) = - 6 x$

$f (x + h) = - 6 x - 6 h$

$f (x) = - 6 x$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{- 6 x - 6 h - (- 6 x)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 6$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{- 6 x - 6 h + 6 x}{h}$

Step 4.1.2

Add $- 6 x$ and $6 x$ .

$f' (x) = lim_{h \to 0} \frac{- 6 h + 0}{h}$

Step 4.1.3

Add $- 6 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{- 6 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{- 6 h}{h}$

Step 4.2.2

Divide $- 6$ by $1$ .

$f' (x) = lim_{h \to 0} - 6$

Step 5

Evaluate the limit of $- 6$ which is constant as $h$ approaches $0$ .

$- 6$

Step 6

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