Calculus Examples

$f (x) = - 6 x$

Consider the limit definition of the derivative.

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

Find the components of the definition.

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

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

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

Simplify the result.

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Apply the distributive property.

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

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

$- 6 x - 6 h$

Find the components of the definition.

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

$f (x) = - 6 x$

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

$f (x) = - 6 x$

Plug in the components.

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

Simplify.

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Simplify the numerator.

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Multiply $- 6$ by $- 1$ .

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

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

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

Add $- 6 h$ and $0$ .

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $- 6$ by $1$ .

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

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

$- 6$

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