Calculus Examples

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

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 1.1

Find the second derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

Differentiate.

Tap for more steps...

Step 1.1.1.2.1

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

Tap for more steps...

Step 1.1.1.2.1.1

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

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

Step 1.1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

Multiply $2$ by $1$ .

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

Step 1.1.1.2.6

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

$\frac{{(x + 4)}^{2} (2 + 0) - (2 x + 2) \frac{d}{d x} [{(x + 4)}^{2}]}{{(x + 4)}^{4}}$

Step 1.1.1.2.7

Simplify the expression.

Tap for more steps...

Step 1.1.1.2.7.1

Add $2$ and $0$ .

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

Step 1.1.1.2.7.2

Move $2$ to the left of ${(x + 4)}^{2}$ .

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

Step 1.1.1.3

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 $g (x) = x + 4$ .

Tap for more steps...

Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

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

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

Step 1.1.1.4

Differentiate.

Tap for more steps...

Step 1.1.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.4.2

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

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

Step 1.1.1.4.3

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

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

Step 1.1.1.4.4

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

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

Step 1.1.1.4.5

Simplify the expression.

Tap for more steps...

Step 1.1.1.4.5.1

Add $1$ and $0$ .

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

Step 1.1.1.4.5.2

Multiply $x + 4$ by $1$ .

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

Step 1.1.1.5

Simplify.

Tap for more steps...

Step 1.1.1.5.1

Apply the distributive property.

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

Step 1.1.1.5.2

Simplify the numerator.

Tap for more steps...

Step 1.1.1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

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

Step 1.1.1.5.2.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.5.2.1.2.1

Apply the distributive property.

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

Step 1.1.1.5.2.1.2.2

Apply the distributive property.

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

Step 1.1.1.5.2.1.2.3

Apply the distributive property.

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

Step 1.1.1.5.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.5.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.1.5.2.1.3.1.2

Move $4$ to the left of $x$ .

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

Step 1.1.1.5.2.1.3.1.3

Multiply $4$ by $4$ .

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

Step 1.1.1.5.2.1.3.2

Add $4 x$ and $4 x$ .

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

Step 1.1.1.5.2.1.4

Apply the distributive property.

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

Step 1.1.1.5.2.1.5

Simplify.

Tap for more steps...

Step 1.1.1.5.2.1.5.1

Multiply $8$ by $2$ .

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

Step 1.1.1.5.2.1.5.2

Multiply $2$ by $16$ .

$\frac{2 x^{2} + 16 x + 32 + (- 2 (2 x) - 2 \cdot 2) (x + 4)}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.6

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.6.1

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} + 16 x + 32 + (- 4 x - 2 \cdot 2) (x + 4)}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.6.2

Multiply $- 2$ by $2$ .

$\frac{2 x^{2} + 16 x + 32 + (- 4 x - 4) (x + 4)}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.7

Expand $(- 4 x - 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.5.2.1.7.1

Apply the distributive property.

$\frac{2 x^{2} + 16 x + 32 - 4 x (x + 4) - 4 (x + 4)}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.7.2

Apply the distributive property.

$\frac{2 x^{2} + 16 x + 32 - 4 x \cdot x - 4 x \cdot 4 - 4 (x + 4)}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.7.3

Apply the distributive property.

$\frac{2 x^{2} + 16 x + 32 - 4 x \cdot x - 4 x \cdot 4 - 4 x - 4 \cdot 4}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.8

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.5.2.1.8.1

Simplify each term.

Tap for more steps...

Step 1.1.1.5.2.1.8.1.1

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

Tap for more steps...

Step 1.1.1.5.2.1.8.1.1.1

Move $x$ .

$\frac{2 x^{2} + 16 x + 32 - 4 (x \cdot x) - 4 x \cdot 4 - 4 x - 4 \cdot 4}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.8.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 16 x + 32 - 4 x^{2} - 4 x \cdot 4 - 4 x - 4 \cdot 4}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.8.1.2

Multiply $4$ by $- 4$ .

$\frac{2 x^{2} + 16 x + 32 - 4 x^{2} - 16 x - 4 x - 4 \cdot 4}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.8.1.3

Multiply $- 4$ by $4$ .

$\frac{2 x^{2} + 16 x + 32 - 4 x^{2} - 16 x - 4 x - 16}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.1.8.2

Subtract $4 x$ from $- 16 x$ .

$\frac{2 x^{2} + 16 x + 32 - 4 x^{2} - 20 x - 16}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.2

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

$\frac{- 2 x^{2} + 16 x + 32 - 20 x - 16}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.3

Subtract $20 x$ from $16 x$ .

$\frac{- 2 x^{2} - 4 x + 32 - 16}{{(x + 4)}^{4}}$

Step 1.1.1.5.2.4

Subtract $16$ from $32$ .

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

Step 1.1.1.5.3

Simplify the numerator.

Tap for more steps...

Step 1.1.1.5.3.1

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

Tap for more steps...

Step 1.1.1.5.3.1.1

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

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

Step 1.1.1.5.3.1.2

Factor $2$ out of $- 4 x$ .

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

Step 1.1.1.5.3.1.3

Factor $2$ out of $16$ .

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

Step 1.1.1.5.3.1.4

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

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

Step 1.1.1.5.3.1.5

Factor $2$ out of $2 (- x^{2} - 2 x) + 2 (8)$ .

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

Step 1.1.1.5.3.2

Factor by grouping.

Tap for more steps...

Step 1.1.1.5.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

Tap for more steps...

Step 1.1.1.5.3.2.1.1

Factor $- 2$ out of $- 2 x$ .

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

Step 1.1.1.5.3.2.1.2

Rewrite $- 2$ as $2$ plus $- 4$

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

Step 1.1.1.5.3.2.1.3

Apply the distributive property.

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

Step 1.1.1.5.3.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.1.1.5.3.2.2.1

Group the first two terms and the last two terms.

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

Step 1.1.1.5.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.1.1.5.3.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

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

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

Step 1.1.1.5.4

Cancel the common factor of $x + 4$ and ${(x + 4)}^{4}$ .

Tap for more steps...

Step 1.1.1.5.4.1

Factor $x + 4$ out of $2 (- x + 2) (x + 4)$ .

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

Step 1.1.1.5.4.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.5.4.2.1

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

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

Step 1.1.1.5.4.2.2

Cancel the common factor.

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

Step 1.1.1.5.4.2.3

Rewrite the expression.

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

Step 1.1.1.5.5

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

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

Step 1.1.1.5.6

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 1.1.1.5.7

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 1.1.1.5.8

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 1.1.1.5.9

Move the negative in front of the fraction.

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

Step 1.1.2

Find the second derivative.

Tap for more steps...

Step 1.1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 (x - 2)}{{(x + 4)}^{3}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x - 2}{{(x + 4)}^{3}}]$ .

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

Step 1.1.2.2

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

Step 1.1.2.3

Differentiate.

Tap for more steps...

Step 1.1.2.3.1

Multiply the exponents in ${({(x + 4)}^{3})}^{2}$ .

Tap for more steps...

Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Multiply $3$ by $2$ .

$- 2 \frac{{(x + 4)}^{3} \frac{d}{d x} [x - 2] - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.3.2

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

$- 2 \frac{{(x + 4)}^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.3.3

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

$- 2 \frac{{(x + 4)}^{3} (1 + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.3.4

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

$- 2 \frac{{(x + 4)}^{3} (1 + 0) - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.3.5

Simplify the expression.

Tap for more steps...

Step 1.1.2.3.5.1

Add $1$ and $0$ .

$- 2 \frac{{(x + 4)}^{3} \cdot 1 - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.3.5.2

Multiply ${(x + 4)}^{3}$ by $1$ .

$- 2 \frac{{(x + 4)}^{3} - (x - 2) \frac{d}{d x} [{(x + 4)}^{3}]}{{(x + 4)}^{6}}$

Step 1.1.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 4$ .

Tap for more steps...

Step 1.1.2.4.1

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

$- 2 \frac{{(x + 4)}^{3} - (x - 2) (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + 4])}{{(x + 4)}^{6}}$

Step 1.1.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$- 2 \frac{{(x + 4)}^{3} - (x - 2) (3 u^{2} \frac{d}{d x} [x + 4])}{{(x + 4)}^{6}}$

Step 1.1.2.4.3

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

$- 2 \frac{{(x + 4)}^{3} - (x - 2) (3 {(x + 4)}^{2} \frac{d}{d x} [x + 4])}{{(x + 4)}^{6}}$

Step 1.1.2.5

Simplify with factoring out.

Tap for more steps...

Step 1.1.2.5.1

Multiply $3$ by $- 1$ .

$- 2 \frac{{(x + 4)}^{3} - 3 (x - 2) ({(x + 4)}^{2} \frac{d}{d x} [x + 4])}{{(x + 4)}^{6}}$

Step 1.1.2.5.2

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

Tap for more steps...

Step 1.1.2.5.2.1

Factor ${(x + 4)}^{2}$ out of ${(x + 4)}^{3}$ .

$- 2 \frac{{(x + 4)}^{2} (x + 4) - 3 (x - 2) ({(x + 4)}^{2} \frac{d}{d x} [x + 4])}{{(x + 4)}^{6}}$

Step 1.1.2.5.2.2

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

$- 2 \frac{{(x + 4)}^{2} (x + 4) + {(x + 4)}^{2} (- 3 (x - 2) (\frac{d}{d x} [x + 4]))}{{(x + 4)}^{6}}$

Step 1.1.2.5.2.3

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

$- 2 \frac{{(x + 4)}^{2} (x + 4 - 3 (x - 2) (\frac{d}{d x} [x + 4]))}{{(x + 4)}^{6}}$

Step 1.1.2.6

Cancel the common factors.

Tap for more steps...

Step 1.1.2.6.1

Factor ${(x + 4)}^{2}$ out of ${(x + 4)}^{6}$ .

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

Step 1.1.2.6.2

Cancel the common factor.

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

Step 1.1.2.6.3

Rewrite the expression.

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Combine fractions.

Tap for more steps...

Step 1.1.2.10.1

Add $1$ and $0$ .

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

Step 1.1.2.10.2

Multiply $- 3$ by $1$ .

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

Step 1.1.2.10.3

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

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

Step 1.1.2.10.4

Move the negative in front of the fraction.

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

Step 1.1.2.11

Simplify.

Tap for more steps...

Step 1.1.2.11.1

Apply the distributive property.

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

Step 1.1.2.11.2

Apply the distributive property.

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

Step 1.1.2.11.3

Simplify the numerator.

Tap for more steps...

Step 1.1.2.11.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.11.3.1.1

Multiply $2$ by $4$ .

$- \frac{2 x + 8 + 2 (- 3 x) + 2 (- 3 \cdot - 2)}{{(x + 4)}^{4}}$

Step 1.1.2.11.3.1.2

Multiply $- 3$ by $2$ .

$- \frac{2 x + 8 - 6 x + 2 (- 3 \cdot - 2)}{{(x + 4)}^{4}}$

Step 1.1.2.11.3.1.3

Multiply $2 (- 3 \cdot - 2)$ .

Tap for more steps...

Step 1.1.2.11.3.1.3.1

Multiply $- 3$ by $- 2$ .

$- \frac{2 x + 8 - 6 x + 2 \cdot 6}{{(x + 4)}^{4}}$

Step 1.1.2.11.3.1.3.2

Multiply $2$ by $6$ .

$- \frac{2 x + 8 - 6 x + 12}{{(x + 4)}^{4}}$

Step 1.1.2.11.3.2

Subtract $6 x$ from $2 x$ .

$- \frac{- 4 x + 8 + 12}{{(x + 4)}^{4}}$

Step 1.1.2.11.3.3

Add $8$ and $12$ .

$- \frac{- 4 x + 20}{{(x + 4)}^{4}}$

Step 1.1.2.11.4

Factor $4$ out of $- 4 x + 20$ .

Tap for more steps...

Step 1.1.2.11.4.1

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

$- \frac{4 (- x) + 20}{{(x + 4)}^{4}}$

Step 1.1.2.11.4.2

Factor $4$ out of $20$ .

$- \frac{4 (- x) + 4 (5)}{{(x + 4)}^{4}}$

Step 1.1.2.11.4.3

Factor $4$ out of $4 (- x) + 4 (5)$ .

$- \frac{4 (- x + 5)}{{(x + 4)}^{4}}$

Step 1.1.2.11.5

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

$- \frac{4 (- (x) + 5)}{{(x + 4)}^{4}}$

Step 1.1.2.11.6

Rewrite $5$ as $- 1 (- 5)$ .

$- \frac{4 (- (x) - 1 (- 5))}{{(x + 4)}^{4}}$

Step 1.1.2.11.7

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$- \frac{4 (- (x - 5))}{{(x + 4)}^{4}}$

Step 1.1.2.11.8

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$- \frac{4 (- 1 (x - 5))}{{(x + 4)}^{4}}$

Step 1.1.2.11.9

Move the negative in front of the fraction.

$- - \frac{4 (x - 5)}{{(x + 4)}^{4}}$

Step 1.1.2.11.10

Multiply $- 1$ by $- 1$ .

$1 \frac{4 (x - 5)}{{(x + 4)}^{4}}$

Step 1.1.2.11.11

Multiply $\frac{4 (x - 5)}{{(x + 4)}^{4}}$ by $1$ .

$f'' (x) = \frac{4 (x - 5)}{{(x + 4)}^{4}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4 (x - 5)}{{(x + 4)}^{4}}$ .

$\frac{4 (x - 5)}{{(x + 4)}^{4}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{4 (x - 5)}{{(x + 4)}^{4}} = 0$ .

Tap for more steps...

Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{4 (x - 5)}{{(x + 4)}^{4}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$4 (x - 5) = 0$

Step 1.2.3

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.3.1

Divide each term in $4 (x - 5) = 0$ by $4$ and simplify.

Tap for more steps...

Step 1.2.3.1.1

Divide each term in $4 (x - 5) = 0$ by $4$ .

$\frac{4 (x - 5)}{4} = \frac{0}{4}$

Step 1.2.3.1.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.1.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{4 (x - 5)}{4} = \frac{0}{4}$

Step 1.2.3.1.2.1.2

Divide $x - 5$ by $1$ .

$x - 5 = \frac{0}{4}$

Step 1.2.3.1.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1.3.1

Divide $0$ by $4$ .

$x - 5 = 0$

Step 1.2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2

Find the domain of $f (x) = \frac{2 x + 2}{{(x + 4)}^{2}}$ .

Tap for more steps...

Step 2.1

Set the denominator in $\frac{2 x + 2}{{(x + 4)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 4)}^{2} = 0$

Step 2.2

Solve for $x$ .

Tap for more steps...

Step 2.2.1

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

$x + 4 = 0$

Step 2.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4) \cup (- 4, \infty)$

Set-Builder Notation:

${x | x \neq - 4}$

Interval Notation:

$(- \infty, - 4) \cup (- 4, \infty)$

Set-Builder Notation:

${x | x \neq - 4}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 4) \cup (- 4, 5) \cup (5, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- 6$ in the expression.

$f'' (- 6) = \frac{4 ((- 6) - 5)}{{((- 6) + 4)}^{4}}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Subtract $5$ from $- 6$ .

$f'' (- 6) = \frac{4 \cdot - 11}{{(- 6 + 4)}^{4}}$

Step 4.2.2

Simplify the denominator.

Tap for more steps...

Step 4.2.2.1

Add $- 6$ and $4$ .

$f'' (- 6) = \frac{4 \cdot - 11}{{(- 2)}^{4}}$

Step 4.2.2.2

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

$f'' (- 6) = \frac{4 \cdot - 11}{16}$

Step 4.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.3.1

Multiply $4$ by $- 11$ .

$f'' (- 6) = \frac{- 44}{16}$

Step 4.2.3.2

Cancel the common factor of $- 44$ and $16$ .

Tap for more steps...

Step 4.2.3.2.1

Factor $4$ out of $- 44$ .

$f'' (- 6) = \frac{4 (- 11)}{16}$

Step 4.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.3.2.2.1

Factor $4$ out of $16$ .

$f'' (- 6) = \frac{4 \cdot - 11}{4 \cdot 4}$

Step 4.2.3.2.2.2

Cancel the common factor.

$f'' (- 6) = \frac{4 \cdot - 11}{4 \cdot 4}$

Step 4.2.3.2.2.3

Rewrite the expression.

$f'' (- 6) = \frac{- 11}{4}$

Step 4.2.3.3

Move the negative in front of the fraction.

$f'' (- 6) = - \frac{11}{4}$

Step 4.2.4

The final answer is $- \frac{11}{4}$ .

$- \frac{11}{4}$

Step 4.3

The graph is concave down on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is negative.

Concave down on $(- \infty, - 4)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- 4, 5)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

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

$f'' (0) = \frac{4 ((0) - 5)}{{((0) + 4)}^{4}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Subtract $5$ from $0$ .

$f'' (0) = \frac{4 \cdot - 5}{{(0 + 4)}^{4}}$

Step 5.2.2

Simplify the denominator.

Tap for more steps...

Step 5.2.2.1

Add $0$ and $4$ .

$f'' (0) = \frac{4 \cdot - 5}{4^{4}}$

Step 5.2.2.2

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

$f'' (0) = \frac{4 \cdot - 5}{256}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.3.1

Multiply $4$ by $- 5$ .

$f'' (0) = \frac{- 20}{256}$

Step 5.2.3.2

Cancel the common factor of $- 20$ and $256$ .

Tap for more steps...

Step 5.2.3.2.1

Factor $4$ out of $- 20$ .

$f'' (0) = \frac{4 (- 5)}{256}$

Step 5.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.2.2.1

Factor $4$ out of $256$ .

$f'' (0) = \frac{4 \cdot - 5}{4 \cdot 64}$

Step 5.2.3.2.2.2

Cancel the common factor.

$f'' (0) = \frac{4 \cdot - 5}{4 \cdot 64}$

Step 5.2.3.2.2.3

Rewrite the expression.

$f'' (0) = \frac{- 5}{64}$

Step 5.2.3.3

Move the negative in front of the fraction.

$f'' (0) = - \frac{5}{64}$

Step 5.2.4

The final answer is $- \frac{5}{64}$ .

$- \frac{5}{64}$

Step 5.3

The graph is concave down on the interval $(- 4, 5)$ because $f'' (0)$ is negative.

Concave down on $(- 4, 5)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(5, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $8$ in the expression.

$f'' (8) = \frac{4 ((8) - 5)}{{((8) + 4)}^{4}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Subtract $5$ from $8$ .

$f'' (8) = \frac{4 \cdot 3}{{(8 + 4)}^{4}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Add $8$ and $4$ .

$f'' (8) = \frac{4 \cdot 3}{12^{4}}$

Step 6.2.2.2

Raise $12$ to the power of $4$ .

$f'' (8) = \frac{4 \cdot 3}{20736}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.2.3.1

Multiply $4$ by $3$ .

$f'' (8) = \frac{12}{20736}$

Step 6.2.3.2

Cancel the common factor of $12$ and $20736$ .

Tap for more steps...

Step 6.2.3.2.1

Factor $12$ out of $12$ .

$f'' (8) = \frac{12 (1)}{20736}$

Step 6.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.3.2.2.1

Factor $12$ out of $20736$ .

$f'' (8) = \frac{12 \cdot 1}{12 \cdot 1728}$

Step 6.2.3.2.2.2

Cancel the common factor.

$f'' (8) = \frac{12 \cdot 1}{12 \cdot 1728}$

Step 6.2.3.2.2.3

Rewrite the expression.

$f'' (8) = \frac{1}{1728}$

Step 6.2.4

The final answer is $\frac{1}{1728}$ .

$\frac{1}{1728}$

Step 6.3

The graph is concave up on the interval $(5, \infty)$ because $f'' (8)$ is positive.

Concave up on $(5, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 4)$ since $f'' (x)$ is negative

Concave down on $(- 4, 5)$ since $f'' (x)$ is negative

Concave up on $(5, \infty)$ since $f'' (x)$ is positive

Step 8