Calculus Examples

$f (x) = \sec (x)$ , $x = \frac{π}{4}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{4}$ .

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Step 1.1

Substitute $\frac{π}{4}$ in for $x$ .

$y = \sec (\frac{π}{4})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \sec (\frac{π}{4})$

Step 1.2.2

Simplify $\sec (\frac{π}{4})$ .

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Step 1.2.2.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$y = \frac{2}{\sqrt{2}}$

Step 1.2.2.2

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

$y = \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2.2.3

Combine and simplify the denominator.

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Step 1.2.2.3.1

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

$y = \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.2.3.4

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

$y = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.2.3.5

Add $1$ and $1$ .

$y = \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.2.3.6

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

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Step 1.2.2.3.6.1

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

$y = \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.2.3.6.2

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

$y = \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.2.3.6.3

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

$y = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.2.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.2.3.6.4.1

Cancel the common factor.

$y = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.2.3.6.4.2

Rewrite the expression.

$y = \frac{2 \sqrt{2}}{2^{1}}$

Step 1.2.2.3.6.5

Evaluate the exponent.

$y = \frac{2 \sqrt{2}}{2}$

Step 1.2.2.4

Cancel the common factor of $2$ .

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Step 1.2.2.4.1

Cancel the common factor.

$y = \frac{2 \sqrt{2}}{2}$

Step 1.2.2.4.2

Divide $\sqrt{2}$ by $1$ .

$y = \sqrt{2}$

Step 2

Find the first derivative and evaluate at $x = \frac{π}{4}$ and $y = \sqrt{2}$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

Evaluate the derivative at $x = \frac{π}{4}$ .

$\sec (\frac{π}{4}) \tan (\frac{π}{4})$

Step 2.3

Simplify.

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Step 2.3.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} \tan (\frac{π}{4})$

Step 2.3.2

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

$\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \tan (\frac{π}{4})$

Step 2.3.3

Combine and simplify the denominator.

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Step 2.3.3.1

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

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} \tan (\frac{π}{4})$

Step 2.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \tan (\frac{π}{4})$

Step 2.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \tan (\frac{π}{4})$

Step 2.3.3.4

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \tan (\frac{π}{4})$

Step 2.3.3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} \tan (\frac{π}{4})$

Step 2.3.3.6

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

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Step 2.3.3.6.1

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

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \tan (\frac{π}{4})$

Step 2.3.3.6.2

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

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \tan (\frac{π}{4})$

Step 2.3.3.6.3

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

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \tan (\frac{π}{4})$

Step 2.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \tan (\frac{π}{4})$

Step 2.3.3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{2}}{2^{1}} \tan (\frac{π}{4})$

Step 2.3.3.6.5

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2} \tan (\frac{π}{4})$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2} \tan (\frac{π}{4})$

Step 2.3.4.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2} \tan (\frac{π}{4})$

Step 2.3.5

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\sqrt{2} \cdot 1$

Step 2.3.6

Multiply $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $\sqrt{2}$ and a given point $(\frac{π}{4}, \sqrt{2})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\sqrt{2}) = \sqrt{2} \cdot (x - (\frac{π}{4}))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - \sqrt{2} = \sqrt{2} \cdot (x - \frac{π}{4})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\sqrt{2} \cdot (x - \frac{π}{4})$ .

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Step 3.3.1.1

Rewrite.

$y - \sqrt{2} = 0 + 0 + \sqrt{2} \cdot (x - \frac{π}{4})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \sqrt{2} = \sqrt{2} \cdot (x - \frac{π}{4})$

Step 3.3.1.3

Apply the distributive property.

$y - \sqrt{2} = \sqrt{2} x + \sqrt{2} (- \frac{π}{4})$

Step 3.3.1.4

Combine $\sqrt{2}$ and $\frac{π}{4}$ .

$y - \sqrt{2} = \sqrt{2} x - \frac{\sqrt{2} π}{4}$

Step 3.3.2

Add $\sqrt{2}$ to both sides of the equation.

$y = \sqrt{2} x - \frac{\sqrt{2} π}{4} + \sqrt{2}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \sqrt{2} x - \frac{\sqrt{2} π}{4} + \sqrt{2} \cdot \frac{4}{4}$

Step 3.3.3.2

Combine $\sqrt{2}$ and $\frac{4}{4}$ .

$y = \sqrt{2} x - \frac{\sqrt{2} π}{4} + \frac{\sqrt{2} \cdot 4}{4}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = \sqrt{2} x + \frac{- \sqrt{2} π + \sqrt{2} \cdot 4}{4}$

Step 3.3.3.4

Move $4$ to the left of $\sqrt{2}$ .

$y = \sqrt{2} x + \frac{- \sqrt{2} π + 4 \sqrt{2}}{4}$

Step 3.3.3.5

Factor $- 1$ out of $- \sqrt{2} π$ .

$y = \sqrt{2} x + \frac{- (\sqrt{2} π) + 4 \sqrt{2}}{4}$

Step 3.3.3.6

Factor $- 1$ out of $4 \sqrt{2}$ .

$y = \sqrt{2} x + \frac{- (\sqrt{2} π) - (- 4 \sqrt{2})}{4}$

Step 3.3.3.7

Factor $- 1$ out of $- (\sqrt{2} π) - (- 4 \sqrt{2})$ .

$y = \sqrt{2} x + \frac{- (\sqrt{2} π - 4 \sqrt{2})}{4}$

Step 3.3.3.8

Rewrite $- (\sqrt{2} π - 4 \sqrt{2})$ as $- 1 (\sqrt{2} π - 4 \sqrt{2})$ .

$y = \sqrt{2} x + \frac{- 1 (\sqrt{2} π - 4 \sqrt{2})}{4}$

Step 3.3.3.9

Move the negative in front of the fraction.

$y = \sqrt{2} x - \frac{\sqrt{2} π - 4 \sqrt{2}}{4}$

Step 4