Calculus Examples

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

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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Step 1.1.1.1.1

To apply the Chain Rule, set $u$ as $x + 3$ .

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

Step 1.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

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

Step 1.1.1.1.3

Replace all occurrences of $u$ with $x + 3$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 1.1.1.2.2

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

$f' (x) = 4 {(x + 3)}^{3} (1 + \frac{d}{d x} (3))$

Step 1.1.1.2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = 4 {(x + 3)}^{3} (1 + 0)$

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $1$ and $0$ .

$f' (x) = 4 {(x + 3)}^{3} \cdot 1$

Step 1.1.1.2.4.2

Multiply $4$ by $1$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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 $g (x) = x + 3$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

Replace all occurrences of $u$ with $x + 3$ .

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

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

Multiply $3$ by $4$ .

$f'' (x) = 12 ({(x + 3)}^{2} \frac{d}{d x} (x + 3))$

Step 1.1.2.3.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$f'' (x) = 12 {(x + 3)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (3))$

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 $n = 1$ .

$f'' (x) = 12 {(x + 3)}^{2} (1 + \frac{d}{d x} (3))$

Step 1.1.2.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = 12 {(x + 3)}^{2} (1 + 0)$

Step 1.1.2.3.5

Simplify the expression.

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Step 1.1.2.3.5.1

Add $1$ and $0$ .

$f'' (x) = 12 {(x + 3)}^{2} \cdot 1$

Step 1.1.2.3.5.2

Multiply $12$ by $1$ .

$f'' (x) = 12 {(x + 3)}^{2}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 {(x + 3)}^{2}$ .

$12 {(x + 3)}^{2}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $12 {(x + 3)}^{2} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$12 {(x + 3)}^{2} = 0$

Step 1.2.2

Divide each term in $12 {(x + 3)}^{2} = 0$ by $12$ and simplify.

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Step 1.2.2.1

Divide each term in $12 {(x + 3)}^{2} = 0$ by $12$ .

$\frac{12 {(x + 3)}^{2}}{12} = \frac{0}{12}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $12$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{12 {(x + 3)}^{2}}{12} = \frac{0}{12}$

Step 1.2.2.2.1.2

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{0}{12}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $12$ .

${(x + 3)}^{2} = 0$

Step 1.2.3

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.4

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

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

$(- \infty, - 3) \cup (- 3, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 3)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 6) = 12 {((- 6) + 3)}^{2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $- 6$ and $3$ .

$f'' (- 6) = 12 {(- 3)}^{2}$

Step 4.2.2

Raise $- 3$ to the power of $2$ .

$f'' (- 6) = 12 \cdot 9$

Step 4.2.3

Multiply $12$ by $9$ .

$f'' (- 6) = 108$

Step 4.2.4

The final answer is $108$ .

$108$

Step 4.3

The graph is concave up on the interval $(- \infty, - 3)$ because $f'' (- 6)$ is positive.

Concave up on $(- \infty, - 3)$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- 3, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = 12 {((0) + 3)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Add $0$ and $3$ .

$f'' (0) = 12 \cdot 3^{2}$

Step 5.2.2

Raise $3$ to the power of $2$ .

$f'' (0) = 12 \cdot 9$

Step 5.2.3

Multiply $12$ by $9$ .

$f'' (0) = 108$

Step 5.2.4

The final answer is $108$ .

$108$

Step 5.3

The graph is concave up on the interval $(- 3, \infty)$ because $f'' (0)$ is positive.

Concave up on $(- 3, \infty)$ since $f'' (x)$ is positive

Step 6

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

Concave up on $(- \infty, - 3)$ since $f'' (x)$ is positive

Concave up on $(- 3, \infty)$ since $f'' (x)$ is positive

Step 7