Calculus Examples

$f (x) = \frac{6}{x}$

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.

$f (x + h) = \frac{6}{x + h}$

The final answer is $\frac{6}{x + h}$ .

$\frac{6}{x + h}$

Find the components of the definition.

$f (x + h) = \frac{6}{x + h}$

$f (x) = \frac{6}{x}$

$f (x + h) = \frac{6}{x + h}$

$f (x) = \frac{6}{x}$

Plug in the components.

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

Simplify.

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Simplify the numerator.

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To write $\frac{6}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

To write $- \frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

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

Write each expression with a common denominator of $(x + h) x$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{6}{x + h}$ by $\frac{x}{x}$ .

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

Multiply $\frac{6}{x}$ by $\frac{x + h}{x + h}$ .

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

Reorder the factors of $(x + h) x$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{6 x - 6 (x + h)}{x (x + h)}$ in a factored form.

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Factor $6$ out of $6 x - 6 (x + h)$ .

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Factor $6$ out of $6 x$ .

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

Factor $6$ out of $- 6 (x + h)$ .

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

Factor $6$ out of $6 (x) + 6 (- (x + h))$ .

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

Apply the distributive property.

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

Subtract $x$ from $x$ .

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

Subtract $h$ from $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{6 \cdot (- 1 h)}{x (x + h)}}{h}$

Combine exponents.

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Factor out negative.

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

Multiply $6$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} - \frac{6 h}{x (x + h)} \cdot \frac{1}{h}$

Cancel the common factor of $h$ .

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Move the leading negative in $- \frac{6 h}{x (x + h)}$ into the numerator.

$f' (x) = lim_{h \to 0} \frac{- 6 h}{x (x + h)} \cdot \frac{1}{h}$

Factor $h$ out of $- 6 h$ .

$f' (x) = lim_{h \to 0} \frac{h \cdot - 6}{x (x + h)} \cdot \frac{1}{h}$

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h \cdot - 6}{x (x + h)} \cdot \frac{1}{h}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Evaluate the limit.

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Move the term $- 1$ outside of the limit because it is constant with respect to $h$ .

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

Move the term $\frac{6}{x}$ outside of the limit because it is constant with respect to $h$ .

$- \frac{6}{x} lim_{h \to 0} \frac{1}{x + h}$

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$- \frac{6}{x} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} x + h}$

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$- \frac{6}{x} \cdot \frac{1}{lim_{h \to 0} x + h}$

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$- \frac{6}{x} \cdot \frac{1}{lim_{h \to 0} x + lim_{h \to 0} h}$

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

$- \frac{6}{x} \cdot \frac{1}{x + lim_{h \to 0} h}$

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- \frac{6}{x} \cdot \frac{1}{x + 0}$

Simplify the answer.

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Add $x$ and $0$ .

$- \frac{6}{x} \cdot \frac{1}{x}$

Multiply $- \frac{6}{x} \cdot \frac{1}{x}$ .

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Multiply $\frac{1}{x}$ by $\frac{6}{x}$ .

$- \frac{6}{x \cdot x}$

Raise $x$ to the power of $1$ .

$- \frac{6}{x^{1} x}$

Raise $x$ to the power of $1$ .

$- \frac{6}{x^{1} x^{1}}$

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

$- \frac{6}{x^{1 + 1}}$

Add $1$ and $1$ .

$- \frac{6}{x^{2}}$