Calculus Examples

$\frac{17}{x^{2} + 12}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $17$ is constant with respect to $x$ , the derivative of $\frac{17}{x^{2} + 12}$ with respect to $x$ is $17 \frac{d}{d x} [\frac{1}{x^{2} + 12}]$ .

$17 \frac{d}{d x} [\frac{1}{x^{2} + 12}]$

Step 1.1.2

Rewrite $\frac{1}{x^{2} + 12}$ as ${(x^{2} + 12)}^{- 1}$ .

$17 \frac{d}{d x} [{(x^{2} + 12)}^{- 1}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + 12$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 12$ .

$17 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 12])$

Step 1.2.2

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

$17 (- u^{- 2} \frac{d}{d x} [x^{2} + 12])$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} + 12$ .

$17 (- {(x^{2} + 12)}^{- 2} \frac{d}{d x} [x^{2} + 12])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $17$ .

$- 17 ({(x^{2} + 12)}^{- 2} \frac{d}{d x} [x^{2} + 12])$

Step 1.3.2

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

$- 17 {(x^{2} + 12)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [12])$

Step 1.3.3

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

$- 17 {(x^{2} + 12)}^{- 2} (2 x + \frac{d}{d x} [12])$

Step 1.3.4

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$- 17 {(x^{2} + 12)}^{- 2} (2 x + 0)$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $2 x$ and $0$ .

$- 17 {(x^{2} + 12)}^{- 2} (2 x)$

Step 1.3.5.2

Multiply $2$ by $- 17$ .

$- 34 {(x^{2} + 12)}^{- 2} x$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 34 \frac{1}{{(x^{2} + 12)}^{2}} x$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $- 34$ and $\frac{1}{{(x^{2} + 12)}^{2}}$ .

$\frac{- 34}{{(x^{2} + 12)}^{2}} x$

Step 1.4.2.2

Move the negative in front of the fraction.

$- \frac{34}{{(x^{2} + 12)}^{2}} x$

Step 1.4.2.3

Combine $x$ and $\frac{34}{{(x^{2} + 12)}^{2}}$ .

$- \frac{x \cdot 34}{{(x^{2} + 12)}^{2}}$

Step 1.4.2.4

Move $34$ to the left of $x$ .

$f' (x) = - \frac{34 x}{{(x^{2} + 12)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 34$ is constant with respect to $x$ , the derivative of $- \frac{34 x}{{(x^{2} + 12)}^{2}}$ with respect to $x$ is $- 34 \frac{d}{d x} [\frac{x}{{(x^{2} + 12)}^{2}}]$ .

$- 34 \frac{d}{d x} [\frac{x}{{(x^{2} + 12)}^{2}}]$

Step 2.2

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 $f (x) = x$ and $g (x) = {(x^{2} + 12)}^{2}$ .

$- 34 \frac{{(x^{2} + 12)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 12)}^{2}]}{{({(x^{2} + 12)}^{2})}^{2}}$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${({(x^{2} + 12)}^{2})}^{2}$ .

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Step 2.3.1.1

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

$- 34 \frac{{(x^{2} + 12)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 12)}^{2}]}{{(x^{2} + 12)}^{2 \cdot 2}}$

Step 2.3.1.2

Multiply $2$ by $2$ .

$- 34 \frac{{(x^{2} + 12)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 12)}^{2}]}{{(x^{2} + 12)}^{4}}$

Step 2.3.2

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

$- 34 \frac{{(x^{2} + 12)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} + 12)}^{2}]}{{(x^{2} + 12)}^{4}}$

Step 2.3.3

Multiply ${(x^{2} + 12)}^{2}$ by $1$ .

$- 34 \frac{{(x^{2} + 12)}^{2} - x \frac{d}{d x} [{(x^{2} + 12)}^{2}]}{{(x^{2} + 12)}^{4}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 12$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 12$ .

$- 34 \frac{{(x^{2} + 12)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{4}}$

Step 2.4.2

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

$- 34 \frac{{(x^{2} + 12)}^{2} - x (2 u \frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{4}}$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} + 12$ .

$- 34 \frac{{(x^{2} + 12)}^{2} - x (2 (x^{2} + 12) \frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{4}}$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

$- 34 \frac{{(x^{2} + 12)}^{2} - 2 x ((x^{2} + 12) \frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{4}}$

Step 2.5.2

Factor $x^{2} + 12$ out of ${(x^{2} + 12)}^{2} - 2 x ((x^{2} + 12) \frac{d}{d x} [x^{2} + 12])$ .

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Step 2.5.2.1

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

$- 34 \frac{(x^{2} + 12) (x^{2} + 12) - 2 x ((x^{2} + 12) \frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{4}}$

Step 2.5.2.2

Factor $x^{2} + 12$ out of $- 2 x ((x^{2} + 12) \frac{d}{d x} [x^{2} + 12])$ .

$- 34 \frac{(x^{2} + 12) (x^{2} + 12) + (x^{2} + 12) (- 2 x (\frac{d}{d x} [x^{2} + 12]))}{{(x^{2} + 12)}^{4}}$

Step 2.5.2.3

Factor $x^{2} + 12$ out of $(x^{2} + 12) (x^{2} + 12) + (x^{2} + 12) (- 2 x (\frac{d}{d x} [x^{2} + 12]))$ .

$- 34 \frac{(x^{2} + 12) (x^{2} + 12 - 2 x (\frac{d}{d x} [x^{2} + 12]))}{{(x^{2} + 12)}^{4}}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

$- 34 \frac{(x^{2} + 12) (x^{2} + 12 - 2 x (\frac{d}{d x} [x^{2} + 12]))}{(x^{2} + 12) {(x^{2} + 12)}^{3}}$

Step 2.6.2

Cancel the common factor.

$- 34 \frac{(x^{2} + 12) (x^{2} + 12 - 2 x (\frac{d}{d x} [x^{2} + 12]))}{(x^{2} + 12) {(x^{2} + 12)}^{3}}$

Step 2.6.3

Rewrite the expression.

$- 34 \frac{x^{2} + 12 - 2 x (\frac{d}{d x} [x^{2} + 12])}{{(x^{2} + 12)}^{3}}$

Step 2.7

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

$- 34 \frac{x^{2} + 12 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [12])}{{(x^{2} + 12)}^{3}}$

Step 2.8

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

$- 34 \frac{x^{2} + 12 - 2 x (2 x + \frac{d}{d x} [12])}{{(x^{2} + 12)}^{3}}$

Step 2.9

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$- 34 \frac{x^{2} + 12 - 2 x (2 x + 0)}{{(x^{2} + 12)}^{3}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

$- 34 \frac{x^{2} + 12 - 2 x (2 x)}{{(x^{2} + 12)}^{3}}$

Step 2.10.2

Multiply $2$ by $- 2$ .

$- 34 \frac{x^{2} + 12 - 4 x \cdot x}{{(x^{2} + 12)}^{3}}$

Step 2.11

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

$- 34 \frac{x^{2} + 12 - 4 (x^{1} x)}{{(x^{2} + 12)}^{3}}$

Step 2.12

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

$- 34 \frac{x^{2} + 12 - 4 (x^{1} x^{1})}{{(x^{2} + 12)}^{3}}$

Step 2.13

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

$- 34 \frac{x^{2} + 12 - 4 x^{1 + 1}}{{(x^{2} + 12)}^{3}}$

Step 2.14

Add $1$ and $1$ .

$- 34 \frac{x^{2} + 12 - 4 x^{2}}{{(x^{2} + 12)}^{3}}$

Step 2.15

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

$- 34 \frac{- 3 x^{2} + 12}{{(x^{2} + 12)}^{3}}$

Step 2.16

Combine $- 34$ and $\frac{- 3 x^{2} + 12}{{(x^{2} + 12)}^{3}}$ .

$\frac{- 34 (- 3 x^{2} + 12)}{{(x^{2} + 12)}^{3}}$

Step 2.17

Move the negative in front of the fraction.

$- \frac{34 (- 3 x^{2} + 12)}{{(x^{2} + 12)}^{3}}$

Step 2.18

Simplify.

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Step 2.18.1

Apply the distributive property.

$- \frac{34 (- 3 x^{2}) + 34 \cdot 12}{{(x^{2} + 12)}^{3}}$

Step 2.18.2

Simplify each term.

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Step 2.18.2.1

Multiply $- 3$ by $34$ .

$- \frac{- 102 x^{2} + 34 \cdot 12}{{(x^{2} + 12)}^{3}}$

Step 2.18.2.2

Multiply $34$ by $12$ .

$f'' (x) = - \frac{- 102 x^{2} + 408}{{(x^{2} + 12)}^{3}}$