Calculus Examples

$f (x) = \frac{4 x^{2}}{x + 2}$

Step 1

Find the derivative.

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Step 1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x^{2}}{x + 2}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{2}}{x + 2}]$ .

$4 \frac{d}{d x} [\frac{x^{2}}{x + 2}]$

Step 1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x + 2$ .

$4 \frac{(x + 2) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 \frac{(x + 2) (2 x) - x^{2} \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3.2

Move $2$ to the left of $x + 2$ .

$4 \frac{2 \cdot (x + 2) x - x^{2} \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$4 \frac{2 (x + 2) x - x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \frac{2 (x + 2) x - x^{2} (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$4 \frac{2 (x + 2) x - x^{2} (1 + 0)}{{(x + 2)}^{2}}$

Step 1.3.6

Combine fractions.

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Step 1.3.6.1

Add $1$ and $0$ .

$4 \frac{2 (x + 2) x - x^{2} \cdot 1}{{(x + 2)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

$4 \frac{2 (x + 2) x - x^{2}}{{(x + 2)}^{2}}$

Step 1.3.6.3

Combine $4$ and $\frac{2 (x + 2) x - x^{2}}{{(x + 2)}^{2}}$ .

$\frac{4 (2 (x + 2) x - x^{2})}{{(x + 2)}^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$\frac{4 ((2 x + 2 \cdot 2) x - x^{2})}{{(x + 2)}^{2}}$

Step 1.4.2

Apply the distributive property.

$\frac{4 (2 x \cdot x + 2 \cdot 2 x - x^{2})}{{(x + 2)}^{2}}$

Step 1.4.3

Apply the distributive property.

$\frac{4 (2 x \cdot x) + 4 (2 \cdot 2 x) + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Simplify each term.

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Step 1.4.4.1.1

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

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Step 1.4.4.1.1.1

Move $x$ .

$\frac{4 (2 (x \cdot x)) + 4 (2 \cdot 2 x) + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4.1.1.2

Multiply $x$ by $x$ .

$\frac{4 (2 x^{2}) + 4 (2 \cdot 2 x) + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4.1.2

Multiply $2$ by $4$ .

$\frac{8 x^{2} + 4 (2 \cdot 2 x) + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4.1.3

Multiply $2$ by $2$ .

$\frac{8 x^{2} + 4 (4 x) + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4.1.4

Multiply $4$ by $4$ .

$\frac{8 x^{2} + 16 x + 4 (- x^{2})}{{(x + 2)}^{2}}$

Step 1.4.4.1.5

Multiply $- 1$ by $4$ .

$\frac{8 x^{2} + 16 x - 4 x^{2}}{{(x + 2)}^{2}}$

Step 1.4.4.2

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{4 x^{2} + 16 x}{{(x + 2)}^{2}}$

Step 1.4.5

Factor $4 x$ out of $4 x^{2} + 16 x$ .

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Step 1.4.5.1

Factor $4 x$ out of $4 x^{2}$ .

$\frac{4 x (x) + 16 x}{{(x + 2)}^{2}}$

Step 1.4.5.2

Factor $4 x$ out of $16 x$ .

$\frac{4 x (x) + 4 x (4)}{{(x + 2)}^{2}}$

Step 1.4.5.3

Factor $4 x$ out of $4 x (x) + 4 x (4)$ .

$\frac{4 x (x + 4)}{{(x + 2)}^{2}}$

Step 2

Set the derivative equal to $0$ then solve the equation $\frac{4 x (x + 4)}{{(x + 2)}^{2}} = 0$ .

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Step 2.1

Set the numerator equal to zero.

$4 x (x + 4) = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 4 = 0$

Step 2.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.2.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.2.4

The final solution is all the values that make $4 x (x + 4) = 0$ true.

$x = 0, - 4$

Step 3

Solve the original function $f (x) = \frac{4 x^{2}}{x + 2}$ at $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{4 {(0)}^{2}}{(0) + 2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Cancel the common factor of $4$ and $(0) + 2$ .

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Step 3.2.1.1

Factor $2$ out of $4 {(0)}^{2}$ .

$f (0) = \frac{2 (2 {(0)}^{2})}{(0) + 2}$

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $2$ out of $0$ .

$f (0) = \frac{2 (2 {(0)}^{2})}{2 (0) + 2}$

Step 3.2.1.2.2

Factor $2$ out of $2$ .

$f (0) = \frac{2 (2 {(0)}^{2})}{2 \cdot 0 + 2 \cdot 1}$

Step 3.2.1.2.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot 1$ .

$f (0) = \frac{2 (2 {(0)}^{2})}{2 \cdot (0 + 1)}$

Step 3.2.1.2.4

Cancel the common factor.

$f (0) = \frac{2 (2 {(0)}^{2})}{2 \cdot (0 + 1)}$

Step 3.2.1.2.5

Rewrite the expression.

$f (0) = \frac{2 {(0)}^{2}}{0 + 1}$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{2 \cdot 0}{0 + 1}$

Step 3.2.2.2

Add $0$ and $1$ .

$f (0) = \frac{2 \cdot 0}{1}$

Step 3.2.2.3

Multiply $2$ by $0$ .

$f (0) = \frac{0}{1}$

Step 3.2.2.4

Divide $0$ by $1$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 4

Solve the original function $f (x) = \frac{4 x^{2}}{x + 2}$ at $x = - 4$ .

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Step 4.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{4 {(- 4)}^{2}}{(- 4) + 2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Cancel the common factor of $4$ and $(- 4) + 2$ .

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Step 4.2.1.1

Factor $2$ out of $4 {(- 4)}^{2}$ .

$f (- 4) = \frac{2 (2 {(- 4)}^{2})}{(- 4) + 2}$

Step 4.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1

Factor $2$ out of $- 4$ .

$f (- 4) = \frac{2 (2 {(- 4)}^{2})}{2 \cdot - 2 + 2}$

Step 4.2.1.2.2

Factor $2$ out of $2$ .

$f (- 4) = \frac{2 (2 {(- 4)}^{2})}{2 \cdot - 2 + 2 (1)}$

Step 4.2.1.2.3

Factor $2$ out of $2 \cdot - 2 + 2 (1)$ .

$f (- 4) = \frac{2 (2 {(- 4)}^{2})}{2 \cdot (- 2 + 1)}$

Step 4.2.1.2.4

Cancel the common factor.

$f (- 4) = \frac{2 (2 {(- 4)}^{2})}{2 \cdot (- 2 + 1)}$

Step 4.2.1.2.5

Rewrite the expression.

$f (- 4) = \frac{2 {(- 4)}^{2}}{- 2 + 1}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = \frac{2 \cdot 16}{- 2 + 1}$

Step 4.2.2.2

Add $- 2$ and $1$ .

$f (- 4) = \frac{2 \cdot 16}{- 1}$

Step 4.2.2.3

Multiply $2$ by $16$ .

$f (- 4) = \frac{32}{- 1}$

Step 4.2.2.4

Divide $32$ by $- 1$ .

$f (- 4) = - 32$

Step 4.2.3

The final answer is $- 32$ .

$- 32$

Step 5

The horizontal tangent lines on function $f (x) = \frac{4 x^{2}}{x + 2}$ are $y = 0, y = - 32$ .

$y = 0, y = - 32$

Step 6