Calculus Examples

${(\frac{x}{12})}^{15}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{15}$ and $g (x) = \frac{x}{12}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{12}$ .

$\frac{d}{d u} [u^{15}] \frac{d}{d x} [\frac{x}{12}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 15$ .

$15 u^{14} \frac{d}{d x} [\frac{x}{12}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{12}$ .

$15 {(\frac{x}{12})}^{14} \frac{d}{d x} [\frac{x}{12}]$

Step 2

Differentiate.

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Step 2.1

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

$15 {(\frac{x}{12})}^{14} (\frac{1}{12} \frac{d}{d x} [x])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $\frac{1}{12}$ and $15$ .

$\frac{15}{12} {(\frac{x}{12})}^{14} \frac{d}{d x} [x]$

Step 2.2.2

Cancel the common factor of $15$ and $12$ .

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Step 2.2.2.1

Factor $3$ out of $15$ .

$\frac{3 (5)}{12} {(\frac{x}{12})}^{14} \frac{d}{d x} [x]$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 5}{3 \cdot 4} {(\frac{x}{12})}^{14} \frac{d}{d x} [x]$

Step 2.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot 5}{3 \cdot 4} {(\frac{x}{12})}^{14} \frac{d}{d x} [x]$

Step 2.2.2.2.3

Rewrite the expression.

$\frac{5}{4} {(\frac{x}{12})}^{14} \frac{d}{d x} [x]$

Step 2.3

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

$\frac{5}{4} {(\frac{x}{12})}^{14} \cdot 1$

Step 2.4

Multiply $\frac{5}{4}$ by $1$ .

$\frac{5}{4} {(\frac{x}{12})}^{14}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{12}$ .

$\frac{5}{4} \cdot \frac{x^{14}}{12^{14}}$

Step 3.2

Combine terms.

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Step 3.2.1

Raise $12$ to the power of $14$ .

$\frac{5}{4} \cdot \frac{x^{14}}{1283918464548860}$

Step 3.2.2

Multiply $\frac{5}{4}$ by $\frac{x^{14}}{1283918464548860}$ .

$\frac{5 x^{14}}{4 \cdot 1283918464548860}$

Step 3.2.3

Multiply $4$ by $1283918464548860$ .

$\frac{5 x^{14}}{5135673858195440}$

Step 3.2.4

Cancel the common factor of $5$ and $5135673858195440$ .

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Step 3.2.4.1

Factor $5$ out of $5 x^{14}$ .

$\frac{5 (x^{14})}{5135673858195440}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $5$ out of $5135673858195440$ .

$\frac{5 x^{14}}{5 \cdot 1027134771639088}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{5 x^{14}}{5 \cdot 1027134771639088}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{x^{14}}{1027134771639088}$