Calculus Examples

$f (x) = 2 x^{3} + 36 x^{2} + 192 x + 7$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{3} + 36 x^{2} + 192 x + 7$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.2.2

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

$2 (3 x^{2}) + \frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.3

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 1.3.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$6 x^{2} + 36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.3.2

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

$6 x^{2} + 36 (2 x) + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.3.3

Multiply $2$ by $36$ .

$6 x^{2} + 72 x + \frac{d}{d x} [192 x] + \frac{d}{d x} [7]$

Step 1.4

Evaluate $\frac{d}{d x} [192 x]$ .

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Step 1.4.1

Since $192$ is constant with respect to $x$ , the derivative of $192 x$ with respect to $x$ is $192 \frac{d}{d x} [x]$ .

$6 x^{2} + 72 x + 192 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 1.4.2

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

$6 x^{2} + 72 x + 192 \cdot 1 + \frac{d}{d x} [7]$

Step 1.4.3

Multiply $192$ by $1$ .

$6 x^{2} + 72 x + 192 + \frac{d}{d x} [7]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$6 x^{2} + 72 x + 192 + 0$

Step 1.5.2

Add $6 x^{2} + 72 x + 192$ and $0$ .

$6 x^{2} + 72 x + 192$

Step 2

Set the derivative equal to $0$ then solve the equation $6 x^{2} + 72 x + 192 = 0$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $6$ out of $6 x^{2} + 72 x + 192$ .

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Step 2.1.1.1

Factor $6$ out of $6 x^{2}$ .

$6 (x^{2}) + 72 x + 192 = 0$

Step 2.1.1.2

Factor $6$ out of $72 x$ .

$6 (x^{2}) + 6 (12 x) + 192 = 0$

Step 2.1.1.3

Factor $6$ out of $192$ .

$6 x^{2} + 6 (12 x) + 6 \cdot 32 = 0$

Step 2.1.1.4

Factor $6$ out of $6 x^{2} + 6 (12 x)$ .

$6 (x^{2} + 12 x) + 6 \cdot 32 = 0$

Step 2.1.1.5

Factor $6$ out of $6 (x^{2} + 12 x) + 6 \cdot 32$ .

$6 (x^{2} + 12 x + 32) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $x^{2} + 12 x + 32$ using the AC method.

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Step 2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $12$ .

$4, 8$

Step 2.1.2.1.2

Write the factored form using these integers.

$6 ((x + 4) (x + 8)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$6 (x + 4) (x + 8) = 0$

Step 2.2

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

$x + 4 = 0$

$x + 8 = 0$

Step 2.3

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

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Step 2.3.1

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

$x + 4 = 0$

Step 2.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.4

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 2.4.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.5

The final solution is all the values that make $6 (x + 4) (x + 8) = 0$ true.

$x = - 4, - 8$

Step 3

Solve the original function $f (x) = 2 x^{3} + 36 x^{2} + 192 x + 7$ at $x = - 4$ .

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Step 3.1

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

$f (- 4) = 2 {(- 4)}^{3} + 36 {(- 4)}^{2} + 192 (- 4) + 7$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 4$ to the power of $3$ .

$f (- 4) = 2 \cdot - 64 + 36 {(- 4)}^{2} + 192 (- 4) + 7$

Step 3.2.1.2

Multiply $2$ by $- 64$ .

$f (- 4) = - 128 + 36 {(- 4)}^{2} + 192 (- 4) + 7$

Step 3.2.1.3

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

$f (- 4) = - 128 + 36 \cdot 16 + 192 (- 4) + 7$

Step 3.2.1.4

Multiply $36$ by $16$ .

$f (- 4) = - 128 + 576 + 192 (- 4) + 7$

Step 3.2.1.5

Multiply $192$ by $- 4$ .

$f (- 4) = - 128 + 576 - 768 + 7$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 128$ and $576$ .

$f (- 4) = 448 - 768 + 7$

Step 3.2.2.2

Subtract $768$ from $448$ .

$f (- 4) = - 320 + 7$

Step 3.2.2.3

Add $- 320$ and $7$ .

$f (- 4) = - 313$

Step 3.2.3

The final answer is $- 313$ .

$- 313$

Step 4

Solve the original function $f (x) = 2 x^{3} + 36 x^{2} + 192 x + 7$ at $x = - 8$ .

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Step 4.1

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

$f (- 8) = 2 {(- 8)}^{3} + 36 {(- 8)}^{2} + 192 (- 8) + 7$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 8$ to the power of $3$ .

$f (- 8) = 2 \cdot - 512 + 36 {(- 8)}^{2} + 192 (- 8) + 7$

Step 4.2.1.2

Multiply $2$ by $- 512$ .

$f (- 8) = - 1024 + 36 {(- 8)}^{2} + 192 (- 8) + 7$

Step 4.2.1.3

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

$f (- 8) = - 1024 + 36 \cdot 64 + 192 (- 8) + 7$

Step 4.2.1.4

Multiply $36$ by $64$ .

$f (- 8) = - 1024 + 2304 + 192 (- 8) + 7$

Step 4.2.1.5

Multiply $192$ by $- 8$ .

$f (- 8) = - 1024 + 2304 - 1536 + 7$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 1024$ and $2304$ .

$f (- 8) = 1280 - 1536 + 7$

Step 4.2.2.2

Subtract $1536$ from $1280$ .

$f (- 8) = - 256 + 7$

Step 4.2.2.3

Add $- 256$ and $7$ .

$f (- 8) = - 249$

Step 4.2.3

The final answer is $- 249$ .

$- 249$

Step 5

The horizontal tangent lines on function $f (x) = 2 x^{3} + 36 x^{2} + 192 x + 7$ are $y = - 313, y = - 249$ .

$y = - 313, y = - 249$

Step 6