Calculus Examples

$f (x) = {(x^{2} + 9)}^{6}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{2} + 9$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} + 9$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [x^{2} + 9]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} + 9$ .

$6 {(x^{2} + 9)}^{5} \frac{d}{d x} [x^{2} + 9]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$6 {(x^{2} + 9)}^{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])$

Step 1.2.2

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

$6 {(x^{2} + 9)}^{5} (2 x + \frac{d}{d x} [9])$

Step 1.2.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$6 {(x^{2} + 9)}^{5} (2 x + 0)$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$6 {(x^{2} + 9)}^{5} (2 x)$

Step 1.2.4.2

Multiply $2$ by $6$ .

$12 {(x^{2} + 9)}^{5} x$

Step 1.2.4.3

Reorder the factors of $12 {(x^{2} + 9)}^{5} x$ .

$f' (x) = 12 x {(x^{2} + 9)}^{5}$

Step 2

Find the second derivative.

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x {(x^{2} + 9)}^{5}$ with respect to $x$ is $12 \frac{d}{d x} [x {(x^{2} + 9)}^{5}]$ .

$12 \frac{d}{d x} [x {(x^{2} + 9)}^{5}]$

Step 2.2

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 $g (x) = {(x^{2} + 9)}^{5}$ .

$12 (x \frac{d}{d x} [{(x^{2} + 9)}^{5}] + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} + 9$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2} + 9$ .

$12 (x (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{2} + 9]) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$12 (x (5 u^{4} \frac{d}{d x} [x^{2} + 9]) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.3.3

Replace all occurrences of $u$ with $x^{2} + 9$ .

$12 (x (5 {(x^{2} + 9)}^{4} \frac{d}{d x} [x^{2} + 9]) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$12 (x (5 {(x^{2} + 9)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.4.2

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

$12 (x (5 {(x^{2} + 9)}^{4} (2 x + \frac{d}{d x} [9])) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.4.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$12 (x (5 {(x^{2} + 9)}^{4} (2 x + 0)) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $2 x$ and $0$ .

$12 (x (5 {(x^{2} + 9)}^{4} (2 x)) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.4.4.2

Multiply $2$ by $5$ .

$12 (x (10 {(x^{2} + 9)}^{4} x) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.5

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

$12 (x^{1} x (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.6

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

$12 (x^{1} x^{1} (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.7

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

$12 (x^{1 + 1} (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.8

Add $1$ and $1$ .

$12 (x^{2} (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5} \frac{d}{d x} [x])$

Step 2.9

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

$12 (x^{2} (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5} \cdot 1)$

Step 2.10

Multiply ${(x^{2} + 9)}^{5}$ by $1$ .

$12 (x^{2} (10 {(x^{2} + 9)}^{4}) + {(x^{2} + 9)}^{5})$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$12 (x^{2} (10 {(x^{2} + 9)}^{4})) + 12 {(x^{2} + 9)}^{5}$

Step 2.11.2

Multiply $10$ by $12$ .

$120 (x^{2} ({(x^{2} + 9)}^{4})) + 12 {(x^{2} + 9)}^{5}$

Step 2.11.3

Factor $12 {(x^{2} + 9)}^{4}$ out of $120 x^{2} {(x^{2} + 9)}^{4} + 12 {(x^{2} + 9)}^{5}$ .

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Step 2.11.3.1

Factor $12 {(x^{2} + 9)}^{4}$ out of $120 x^{2} {(x^{2} + 9)}^{4}$ .

$12 {(x^{2} + 9)}^{4} (10 x^{2}) + 12 {(x^{2} + 9)}^{5}$

Step 2.11.3.2

Factor $12 {(x^{2} + 9)}^{4}$ out of $12 {(x^{2} + 9)}^{5}$ .

$12 {(x^{2} + 9)}^{4} (10 x^{2}) + 12 {(x^{2} + 9)}^{4} (x^{2} + 9)$

Step 2.11.3.3

Factor $12 {(x^{2} + 9)}^{4}$ out of $12 {(x^{2} + 9)}^{4} (10 x^{2}) + 12 {(x^{2} + 9)}^{4} (x^{2} + 9)$ .

$12 {(x^{2} + 9)}^{4} (10 x^{2} + x^{2} + 9)$

Step 2.11.4

Add $10 x^{2}$ and $x^{2}$ .

$f'' (x) = 12 {(x^{2} + 9)}^{4} (11 x^{2} + 9)$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $12 {(x^{2} + 9)}^{4} (11 x^{2} + 9)$ .

$12 {(x^{2} + 9)}^{4} (11 x^{2} + 9)$