Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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 $f (x) = x$ and $g (x) = {(x + 4)}^{2}$ .

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

Step 1.1.3

Differentiate using the Power Rule.

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Step 1.1.3.1

Multiply the exponents in ${({(x + 4)}^{2})}^{2}$ .

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Step 1.1.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.3.1.2

Multiply $2$ by $2$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply ${(x + 4)}^{2}$ by $1$ .

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

Step 1.1.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x + 4$ .

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Step 1.1.4.1

To apply the Chain Rule, set $u$ as $x + 4$ .

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

Step 1.1.4.2

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

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

Step 1.1.4.3

Replace all occurrences of $u$ with $x + 4$ .

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

Step 1.1.5

Simplify with factoring out.

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Step 1.1.5.1

Multiply $2$ by $- 1$ .

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

Step 1.1.5.2

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

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Step 1.1.5.2.1

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

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

Step 1.1.5.2.2

Factor $x + 4$ out of $- 2 x ((x + 4) \frac{d}{d x} [x + 4])$ .

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

Step 1.1.5.2.3

Factor $x + 4$ out of $(x + 4) (x + 4) + (x + 4) (- 2 x (\frac{d}{d x} [x + 4]))$ .

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

Step 1.1.6

Cancel the common factors.

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Step 1.1.6.1

Factor $x + 4$ out of ${(x + 4)}^{4}$ .

$4 \frac{(x + 4) (x + 4 - 2 x (\frac{d}{d x} [x + 4]))}{(x + 4) {(x + 4)}^{3}}$

Step 1.1.6.2

Cancel the common factor.

$4 \frac{(x + 4) (x + 4 - 2 x (\frac{d}{d x} [x + 4]))}{(x + 4) {(x + 4)}^{3}}$

Step 1.1.6.3

Rewrite the expression.

$4 \frac{x + 4 - 2 x (\frac{d}{d x} [x + 4])}{{(x + 4)}^{3}}$

Step 1.1.7

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$4 \frac{x + 4 - 2 x (\frac{d}{d x} [x] + \frac{d}{d x} [4])}{{(x + 4)}^{3}}$

Step 1.1.8

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

$4 \frac{x + 4 - 2 x (1 + \frac{d}{d x} [4])}{{(x + 4)}^{3}}$

Step 1.1.9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$4 \frac{x + 4 - 2 x (1 + 0)}{{(x + 4)}^{3}}$

Step 1.1.10

Simplify terms.

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Step 1.1.10.1

Add $1$ and $0$ .

$4 \frac{x + 4 - 2 x \cdot 1}{{(x + 4)}^{3}}$

Step 1.1.10.2

Multiply $- 2$ by $1$ .

$4 \frac{x + 4 - 2 x}{{(x + 4)}^{3}}$

Step 1.1.10.3

Subtract $2 x$ from $x$ .

$4 \frac{- x + 4}{{(x + 4)}^{3}}$

Step 1.1.10.4

Combine $4$ and $\frac{- x + 4}{{(x + 4)}^{3}}$ .

$\frac{4 (- x + 4)}{{(x + 4)}^{3}}$

Step 1.1.11

Simplify.

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Step 1.1.11.1

Apply the distributive property.

$\frac{4 (- x) + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 1.1.11.2

Simplify each term.

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Step 1.1.11.2.1

Multiply $- 1$ by $4$ .

$\frac{- 4 x + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 1.1.11.2.2

Multiply $4$ by $4$ .

$\frac{- 4 x + 16}{{(x + 4)}^{3}}$

Step 1.1.11.3

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

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Step 1.1.11.3.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 16}{{(x + 4)}^{3}}$

Step 1.1.11.3.2

Factor $4$ out of $16$ .

$\frac{4 (- x) + 4 (4)}{{(x + 4)}^{3}}$

Step 1.1.11.3.3

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

$\frac{4 (- x + 4)}{{(x + 4)}^{3}}$

Step 1.1.11.4

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) + 4)}{{(x + 4)}^{3}}$

Step 1.1.11.5

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{4 (- (x) - 1 (- 4))}{{(x + 4)}^{3}}$

Step 1.1.11.6

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{4 (- (x - 4))}{{(x + 4)}^{3}}$

Step 1.1.11.7

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{4 (- 1 (x - 4))}{{(x + 4)}^{3}}$

Step 1.1.11.8

Move the negative in front of the fraction.

$f' (x) = - \frac{4 (x - 4)}{{(x + 4)}^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 (x - 4)}{{(x + 4)}^{3}}$ .

$- \frac{4 (x - 4)}{{(x + 4)}^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4 (x - 4)}{{(x + 4)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4 (x - 4)}{{(x + 4)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 (x - 4) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $4 (x - 4) = 0$ by $4$ and simplify.

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Step 2.3.1.1

Divide each term in $4 (x - 4) = 0$ by $4$ .

$\frac{4 (x - 4)}{4} = \frac{0}{4}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $4$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x - 4)}{4} = \frac{0}{4}$

Step 2.3.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{4}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $4$ .

$x - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{4 (x - 4)}{{(x + 4)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x + 4)}^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x + 4 = 0$

Step 3.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

Evaluate $\frac{4 x}{{(x + 4)}^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 4$ .

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Step 4.1.1

Substitute $4$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $4$ by $4$ .

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

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Add $4$ and $4$ .

$\frac{16}{8^{2}}$

Step 4.1.2.2.2

Raise $8$ to the power of $2$ .

$\frac{16}{64}$

Step 4.1.2.3

Cancel the common factor of $16$ and $64$ .

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Step 4.1.2.3.1

Factor $16$ out of $16$ .

$\frac{16 (1)}{64}$

Step 4.1.2.3.2

Cancel the common factors.

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Step 4.1.2.3.2.1

Factor $16$ out of $64$ .

$\frac{16 \cdot 1}{16 \cdot 4}$

Step 4.1.2.3.2.2

Cancel the common factor.

$\frac{16 \cdot 1}{16 \cdot 4}$

Step 4.1.2.3.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 4.2

Evaluate at $x = - 4$ .

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Step 4.2.1

Substitute $- 4$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Add $- 4$ and $4$ .

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

Step 4.2.2.2

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

$\frac{4 (- 4)}{0}$

Step 4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(4, \frac{1}{4})$

Step 5