Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

The final answer is $\frac{2 (x + h)}{x + h + 3}$ .

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h)}{x + h + 3} - (\frac{2 x}{x + 3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{2 (x + h)}{x + h + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h)}{x + h + 3} \cdot \frac{x + 3}{x + 3} - \frac{2 x}{x + 3}}{h}$

Step 4.1.2

To write $- \frac{2 x}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + h + 3}{x + h + 3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h)}{x + h + 3} \cdot \frac{x + 3}{x + 3} - \frac{2 x}{x + 3} \cdot \frac{x + h + 3}{x + h + 3}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h + 3) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{2 (x + h)}{x + h + 3}$ by $\frac{x + 3}{x + 3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h) (x + 3)}{(x + h + 3) (x + 3)} - \frac{2 x}{x + 3} \cdot \frac{x + h + 3}{x + h + 3}}{h}$

Step 4.1.3.2

Multiply $\frac{2 x}{x + 3}$ by $\frac{x + h + 3}{x + h + 3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h) (x + 3)}{(x + h + 3) (x + 3)} - \frac{2 x (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x + h + 3) (x + 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h) (x + 3)}{(x + 3) (x + h + 3)} - \frac{2 x (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x + h) (x + 3) - 2 x (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5

Rewrite $\frac{2 (x + h) (x + 3) - 2 x (x + h + 3)}{(x + 3) (x + h + 3)}$ in a factored form.

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Step 4.1.5.1

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

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Step 4.1.5.1.1

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

$f' (x) = lim_{h \to 0} \frac{\frac{2 ((x + h) (x + 3)) - 2 x (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.1.2

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

$f' (x) = lim_{h \to 0} \frac{\frac{2 ((x + h) (x + 3)) + 2 (- x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.1.3

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

$f' (x) = lim_{h \to 0} \frac{\frac{2 ((x + h) (x + 3) - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.2

Expand $(x + h) (x + 3)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x (x + 3) + h (x + 3) - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.2.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x \cdot x + x \cdot 3 + h (x + 3) - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x \cdot x + x \cdot 3 + h x + h \cdot 3 - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.3

Simplify each term.

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Step 4.1.5.3.1

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + x \cdot 3 + h x + h \cdot 3 - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.3.2

Move $3$ to the left of $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 \cdot x + h x + h \cdot 3 - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.3.3

Move $3$ to the left of $h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 x + h x + 3 h - x (x + h + 3))}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 x + h x + 3 h - x \cdot x - x h - x \cdot 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.5

Simplify.

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Step 4.1.5.5.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.5.1.1

Move $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 x + h x + 3 h - (x \cdot x) - x h - x \cdot 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.5.1.2

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 x + h x + 3 h - x^{2} - x h - x \cdot 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.5.2

Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (x^{2} + 3 x + h x + 3 h - x^{2} - x h - 3 x)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.6

Subtract $x^{2}$ from $x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (3 x + h x + 3 h + 0 - x h - 3 x)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.7

Add $3 x$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (3 x + h x + 3 h - x h - 3 x)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.8

Subtract $3 x$ from $3 x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (h x + 3 h - x h + 0)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.9

Add $h x + 3 h - x h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (h x + 3 h - x h)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.10

Subtract $x h$ from $h x$ .

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Step 4.1.5.10.1

Reorder $h$ and $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (3 h + x h - x h)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.10.2

Subtract $x h$ from $x h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 (3 h + 0)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.11

Add $3 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 \cdot (3 h)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.12

Multiply $2$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{6 h}{(x + 3) (x + h + 3)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} \frac{6 h}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $6 h$ .

$f' (x) = lim_{h \to 0} \frac{h \cdot 6}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h \cdot 6}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$f' (x) = lim_{h \to 0} \frac{6}{(x + 3) (x + h + 3)}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $\frac{6}{x + 3}$ outside of the limit because it is constant with respect to $h$ .

$\frac{6}{x + 3} lim_{h \to 0} \frac{1}{x + h + 3}$

Step 5.2

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

$\frac{6}{x + 3} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} x + h + 3}$

Step 5.3

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

$\frac{6}{x + 3} \cdot \frac{1}{lim_{h \to 0} x + h + 3}$

Step 5.4

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

$\frac{6}{x + 3} \cdot \frac{1}{lim_{h \to 0} x + lim_{h \to 0} h + lim_{h \to 0} 3}$

Step 5.5

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

$\frac{6}{x + 3} \cdot \frac{1}{x + lim_{h \to 0} h + lim_{h \to 0} 3}$

Step 5.6

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$\frac{6}{x + 3} \cdot \frac{1}{x + lim_{h \to 0} h + 3}$

Step 6

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

$\frac{6}{x + 3} \cdot \frac{1}{x + 0 + 3}$

Step 7

Simplify the answer.

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Step 7.1

Add $x$ and $0$ .

$\frac{6}{x + 3} \cdot \frac{1}{x + 3}$

Step 7.2

Multiply $\frac{6}{x + 3} \cdot \frac{1}{x + 3}$ .

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Step 7.2.1

Multiply $\frac{6}{x + 3}$ by $\frac{1}{x + 3}$ .

$\frac{6}{(x + 3) (x + 3)}$

Step 7.2.2

Raise $x + 3$ to the power of $1$ .

$\frac{6}{{(x + 3)}^{1} (x + 3)}$

Step 7.2.3

Raise $x + 3$ to the power of $1$ .

$\frac{6}{{(x + 3)}^{1} {(x + 3)}^{1}}$

Step 7.2.4

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

$\frac{6}{{(x + 3)}^{1 + 1}}$

Step 7.2.5

Add $1$ and $1$ .

$\frac{6}{{(x + 3)}^{2}}$

Step 8