Calculus Examples

$y = 9 \sec (\frac{1}{3} x)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{d}{d x} [9 \sec (\frac{x}{3})]$

Step 1.2

Since $9$ is constant with respect to $x$ , the derivative of $9 \sec (\frac{x}{3})$ with respect to $x$ is $9 \frac{d}{d x} [\sec (\frac{x}{3})]$ .

$9 \frac{d}{d x} [\sec (\frac{x}{3})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = \frac{x}{3}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x}{3}$ .

$9 (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [\frac{x}{3}])$

Step 2.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$9 (\sec (u) \tan (u) \frac{d}{d x} [\frac{x}{3}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$9 (\sec (\frac{x}{3}) \tan (\frac{x}{3}) \frac{d}{d x} [\frac{x}{3}])$

Step 3

Differentiate.

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Step 3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$9 \sec (\frac{x}{3}) \tan (\frac{x}{3}) (\frac{1}{3} \frac{d}{d x} [x])$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $\frac{1}{3}$ and $9$ .

$\frac{9}{3} \sec (\frac{x}{3}) \tan (\frac{x}{3}) \frac{d}{d x} [x]$

Step 3.2.2

Combine $\frac{9}{3}$ and $\sec (\frac{x}{3})$ .

$\frac{9 \sec (\frac{x}{3})}{3} \tan (\frac{x}{3}) \frac{d}{d x} [x]$

Step 3.2.3

Combine $\frac{9 \sec (\frac{x}{3})}{3}$ and $\tan (\frac{x}{3})$ .

$\frac{9 \sec (\frac{x}{3}) \tan (\frac{x}{3})}{3} \frac{d}{d x} [x]$

Step 3.2.4

Cancel the common factor of $9$ and $3$ .

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Step 3.2.4.1

Factor $3$ out of $9 \sec (\frac{x}{3}) \tan (\frac{x}{3})$ .

$\frac{3 (3 \sec (\frac{x}{3}) \tan (\frac{x}{3}))}{3} \frac{d}{d x} [x]$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (3 \sec (\frac{x}{3}) \tan (\frac{x}{3}))}{3 (1)} \frac{d}{d x} [x]$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{3 (3 \sec (\frac{x}{3}) \tan (\frac{x}{3}))}{3 \cdot 1} \frac{d}{d x} [x]$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{3 \sec (\frac{x}{3}) \tan (\frac{x}{3})}{1} \frac{d}{d x} [x]$

Step 3.2.4.2.4

Divide $3 \sec (\frac{x}{3}) \tan (\frac{x}{3})$ by $1$ .

$3 \sec (\frac{x}{3}) \tan (\frac{x}{3}) \frac{d}{d x} [x]$

Step 3.3

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

$3 \sec (\frac{x}{3}) \tan (\frac{x}{3}) \cdot 1$

Step 3.4

Multiply $3$ by $1$ .

$3 \sec (\frac{x}{3}) \tan (\frac{x}{3})$