Calculus Examples

$(x) (22 - 2 x) (22 - 2 x)$

Step 1

Find the first derivative.

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Step 1.1

Raise $22 - 2 x$ to the power of $1$ .

$\frac{d}{d x} [x ({(22 - 2 x)}^{1} (22 - 2 x))]$

Step 1.2

Raise $22 - 2 x$ to the power of $1$ .

$\frac{d}{d x} [x ({(22 - 2 x)}^{1} {(22 - 2 x)}^{1})]$

Step 1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [x {(22 - 2 x)}^{1 + 1}]$

Step 1.4

Add $1$ and $1$ .

$\frac{d}{d x} [x {(22 - 2 x)}^{2}]$

Step 1.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(22 - 2 x)}^{2}$ .

$x \frac{d}{d x} [{(22 - 2 x)}^{2}] + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.6

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 $g (x) = 22 - 2 x$ .

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Step 1.6.1

To apply the Chain Rule, set $u$ as $22 - 2 x$ .

$x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [22 - 2 x]) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.6.2

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

$x (2 u \frac{d}{d x} [22 - 2 x]) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.6.3

Replace all occurrences of $u$ with $22 - 2 x$ .

$x (2 (22 - 2 x) \frac{d}{d x} [22 - 2 x]) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7

Differentiate.

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Step 1.7.1

By the Sum Rule, the derivative of $22 - 2 x$ with respect to $x$ is $\frac{d}{d x} [22] + \frac{d}{d x} [- 2 x]$ .

$x (2 (22 - 2 x) (\frac{d}{d x} [22] + \frac{d}{d x} [- 2 x])) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.2

Since $22$ is constant with respect to $x$ , the derivative of $22$ with respect to $x$ is $0$ .

$x (2 (22 - 2 x) (0 + \frac{d}{d x} [- 2 x])) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$x (2 (22 - 2 x) \frac{d}{d x} [- 2 x]) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.4

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

$x (2 (22 - 2 x) (- 2 \frac{d}{d x} [x])) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.5

Multiply $- 2$ by $2$ .

$x (- 4 (22 - 2 x) \frac{d}{d x} [x]) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.6

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

$x (- 4 (22 - 2 x) \cdot 1) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.7

Multiply $- 4$ by $1$ .

$x (- 4 (22 - 2 x)) + {(22 - 2 x)}^{2} \frac{d}{d x} [x]$

Step 1.7.8

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

$x (- 4 (22 - 2 x)) + {(22 - 2 x)}^{2} \cdot 1$

Step 1.7.9

Multiply ${(22 - 2 x)}^{2}$ by $1$ .

$x (- 4 (22 - 2 x)) + {(22 - 2 x)}^{2}$

Step 1.8

Simplify.

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Step 1.8.1

Apply the distributive property.

$x (- 4 \cdot 22 - 4 (- 2 x)) + {(22 - 2 x)}^{2}$

Step 1.8.2

Apply the distributive property.

$x (- 4 \cdot 22) + x (- 4 (- 2 x)) + {(22 - 2 x)}^{2}$

Step 1.8.3

Combine terms.

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Step 1.8.3.1

Multiply $- 4$ by $22$ .

$x \cdot - 88 + x (- 4 (- 2 x)) + {(22 - 2 x)}^{2}$

Step 1.8.3.2

Move $- 88$ to the left of $x$ .

$- 88 \cdot x + x (- 4 (- 2 x)) + {(22 - 2 x)}^{2}$

Step 1.8.3.3

Multiply $- 2$ by $- 4$ .

$- 88 x + x (8 x) + {(22 - 2 x)}^{2}$

Step 1.8.3.4

Raise $x$ to the power of $1$ .

$- 88 x + 8 (x^{1} x) + {(22 - 2 x)}^{2}$

Step 1.8.3.5

Raise $x$ to the power of $1$ .

$- 88 x + 8 (x^{1} x^{1}) + {(22 - 2 x)}^{2}$

Step 1.8.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 88 x + 8 x^{1 + 1} + {(22 - 2 x)}^{2}$

Step 1.8.3.7

Add $1$ and $1$ .

$- 88 x + 8 x^{2} + {(22 - 2 x)}^{2}$

Step 1.8.4

Reorder terms.

$8 x^{2} + {(22 - 2 x)}^{2} - 88 x$

Step 1.8.5

Simplify each term.

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Step 1.8.5.1

Rewrite ${(22 - 2 x)}^{2}$ as $(22 - 2 x) (22 - 2 x)$ .

$8 x^{2} + (22 - 2 x) (22 - 2 x) - 88 x$

Step 1.8.5.2

Expand $(22 - 2 x) (22 - 2 x)$ using the FOIL Method.

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Step 1.8.5.2.1

Apply the distributive property.

$8 x^{2} + 22 (22 - 2 x) - 2 x (22 - 2 x) - 88 x$

Step 1.8.5.2.2

Apply the distributive property.

$8 x^{2} + 22 \cdot 22 + 22 (- 2 x) - 2 x (22 - 2 x) - 88 x$

Step 1.8.5.2.3

Apply the distributive property.

$8 x^{2} + 22 \cdot 22 + 22 (- 2 x) - 2 x \cdot 22 - 2 x (- 2 x) - 88 x$

Step 1.8.5.3

Simplify and combine like terms.

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Step 1.8.5.3.1

Simplify each term.

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Step 1.8.5.3.1.1

Multiply $22$ by $22$ .

$8 x^{2} + 484 + 22 (- 2 x) - 2 x \cdot 22 - 2 x (- 2 x) - 88 x$

Step 1.8.5.3.1.2

Multiply $- 2$ by $22$ .

$8 x^{2} + 484 - 44 x - 2 x \cdot 22 - 2 x (- 2 x) - 88 x$

Step 1.8.5.3.1.3

Multiply $22$ by $- 2$ .

$8 x^{2} + 484 - 44 x - 44 x - 2 x (- 2 x) - 88 x$

Step 1.8.5.3.1.4

Rewrite using the commutative property of multiplication.

$8 x^{2} + 484 - 44 x - 44 x - 2 \cdot - 2 x \cdot x - 88 x$

Step 1.8.5.3.1.5

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

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Step 1.8.5.3.1.5.1

Move $x$ .

$8 x^{2} + 484 - 44 x - 44 x - 2 \cdot - 2 (x \cdot x) - 88 x$

Step 1.8.5.3.1.5.2

Multiply $x$ by $x$ .

$8 x^{2} + 484 - 44 x - 44 x - 2 \cdot - 2 x^{2} - 88 x$

Step 1.8.5.3.1.6

Multiply $- 2$ by $- 2$ .

$8 x^{2} + 484 - 44 x - 44 x + 4 x^{2} - 88 x$

Step 1.8.5.3.2

Subtract $44 x$ from $- 44 x$ .

$8 x^{2} + 484 - 88 x + 4 x^{2} - 88 x$

Step 1.8.6

Add $8 x^{2}$ and $4 x^{2}$ .

$12 x^{2} + 484 - 88 x - 88 x$

Step 1.8.7

Subtract $88 x$ from $- 88 x$ .

$f' (x) = 12 x^{2} + 484 - 176 x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $12 x^{2} + 484 - 176 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [484] + \frac{d}{d x} [- 176 x]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [484] + \frac{d}{d x} [- 176 x]$

Step 2.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [484] + \frac{d}{d x} [- 176 x]$

Step 2.2.2

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

$12 (2 x) + \frac{d}{d x} [484] + \frac{d}{d x} [- 176 x]$

Step 2.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [484] + \frac{d}{d x} [- 176 x]$

Step 2.3

Since $484$ is constant with respect to $x$ , the derivative of $484$ with respect to $x$ is $0$ .

$24 x + 0 + \frac{d}{d x} [- 176 x]$

Step 2.4

Evaluate $\frac{d}{d x} [- 176 x]$ .

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Step 2.4.1

Since $- 176$ is constant with respect to $x$ , the derivative of $- 176 x$ with respect to $x$ is $- 176 \frac{d}{d x} [x]$ .

$24 x + 0 - 176 \frac{d}{d x} [x]$

Step 2.4.2

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

$24 x + 0 - 176 \cdot 1$

Step 2.4.3

Multiply $- 176$ by $1$ .

$24 x + 0 - 176$

Step 2.5

Add $24 x$ and $0$ .

$f'' (x) = 24 x - 176$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $24 x - 176$ with respect to $x$ is $\frac{d}{d x} [24 x] + \frac{d}{d x} [- 176]$ .

$\frac{d}{d x} [24 x] + \frac{d}{d x} [- 176]$

Step 3.2

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 3.2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$24 \frac{d}{d x} [x] + \frac{d}{d x} [- 176]$

Step 3.2.2

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

$24 \cdot 1 + \frac{d}{d x} [- 176]$

Step 3.2.3

Multiply $24$ by $1$ .

$24 + \frac{d}{d x} [- 176]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $- 176$ is constant with respect to $x$ , the derivative of $- 176$ with respect to $x$ is $0$ .

$24 + 0$

Step 3.3.2

Add $24$ and $0$ .

$f''' (x) = 24$

Step 4

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$f^{4} (x) = 0$