Calculus Examples

$y = \sec (x)$ , $(\frac{π}{6}, \frac{2 \sqrt{3}}{3})$

Step 1

Find the derivative of the function. To find the slope of the equation tangent to the line, evaluate the derivative at the desired value of $x$ .

$\frac{d}{d x} (\sec (x))$

Step 2

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

$\sec (x) \tan (x)$

Step 3

The derivative of the equation in terms of $y$ can also be represented as $f' (x)$ .

$f' (x) = \sec (x) \tan (x)$

Step 4

Replace the variable $x$ with $\frac{π}{6}$ in the expression.

$f' (\frac{π}{6}) = \sec (\frac{π}{6}) \tan (\frac{π}{6})$

Step 5

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$f' (\frac{π}{6}) = \frac{2}{\sqrt{3}} \cdot \tan (\frac{π}{6})$

Step 6

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f' (\frac{π}{6}) = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot \tan (\frac{π}{6})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \tan (\frac{π}{6})$

Step 7.2

Raise $\sqrt{3}$ to the power of $1$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \tan (\frac{π}{6})$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \tan (\frac{π}{6})$

Step 7.4

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

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \tan (\frac{π}{6})$

Step 7.5

Add $1$ and $1$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \cdot \tan (\frac{π}{6})$

Step 7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \tan (\frac{π}{6})$

Step 7.6.2

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

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \tan (\frac{π}{6})$

Step 7.6.3

Combine $\frac{1}{2}$ and $2$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \tan (\frac{π}{6})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \tan (\frac{π}{6})$

Step 7.6.4.2

Rewrite the expression.

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3} \cdot \tan (\frac{π}{6})$

Step 7.6.5

Evaluate the exponent.

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3} \cdot \tan (\frac{π}{6})$

Step 8

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3}$

Step 9

Multiply $\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3}$ .

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Step 9.1

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{\sqrt{3}}{3}$ .

$f' (\frac{π}{6}) = \frac{2 \sqrt{3} \sqrt{3}}{3 \cdot 3}$

Step 9.2

Raise $\sqrt{3}$ to the power of $1$ .

$f' (\frac{π}{6}) = \frac{2 (\sqrt{3} \sqrt{3})}{3 \cdot 3}$

Step 9.3

Raise $\sqrt{3}$ to the power of $1$ .

$f' (\frac{π}{6}) = \frac{2 (\sqrt{3} \sqrt{3})}{3 \cdot 3}$

Step 9.4

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

$f' (\frac{π}{6}) = \frac{2 {\sqrt{3}}^{1 + 1}}{3 \cdot 3}$

Step 9.5

Add $1$ and $1$ .

$f' (\frac{π}{6}) = \frac{2 {\sqrt{3}}^{2}}{3 \cdot 3}$

Step 9.6

Multiply $3$ by $3$ .

$f' (\frac{π}{6}) = \frac{2 {\sqrt{3}}^{2}}{9}$

Step 10

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f' (\frac{π}{6}) = \frac{2 {(3^{\frac{1}{2}})}^{2}}{9}$

Step 10.2

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

$f' (\frac{π}{6}) = \frac{2 \cdot 3^{\frac{1}{2} \cdot 2}}{9}$

Step 10.3

Combine $\frac{1}{2}$ and $2$ .

$f' (\frac{π}{6}) = \frac{2 \cdot 3^{\frac{2}{2}}}{9}$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$f' (\frac{π}{6}) = \frac{2 \cdot 3^{\frac{2}{2}}}{9}$

Step 10.4.2

Rewrite the expression.

$f' (\frac{π}{6}) = \frac{2 \cdot 3}{9}$

Step 10.5

Evaluate the exponent.

$f' (\frac{π}{6}) = \frac{2 \cdot 3}{9}$

Step 11

Multiply $2$ by $3$ .

$f' (\frac{π}{6}) = \frac{6}{9}$

Step 12

Cancel the common factor of $6$ and $9$ .

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Step 12.1

Factor $3$ out of $6$ .

$f' (\frac{π}{6}) = \frac{3 (2)}{9}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $3$ out of $9$ .

$f' (\frac{π}{6}) = \frac{3 \cdot 2}{3 \cdot 3}$

Step 12.2.2

Cancel the common factor.

$f' (\frac{π}{6}) = \frac{3 \cdot 2}{3 \cdot 3}$

Step 12.2.3

Rewrite the expression.

$f' (\frac{π}{6}) = \frac{2}{3}$

Step 13

The derivative at $(\frac{π}{6}, \frac{2 \sqrt{3}}{3})$ is $\frac{2}{3}$ .

$\frac{2}{3}$