Calculus Examples

$f (x) = \sec (\frac{π}{4})$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [\frac{2}{\sqrt{2}}]$

Step 1.2

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

$\frac{d}{d x} [\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}]$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine and simplify the denominator.

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Step 1.3.1.1

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}]$

Step 1.3.1.2

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}]$

Step 1.3.1.3

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}]$

Step 1.3.1.4

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

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

Step 1.3.1.5

Add $1$ and $1$ .

$\frac{d}{d x} [\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}]$

Step 1.3.1.6

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

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Step 1.3.1.6.1

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}]$

Step 1.3.1.6.2

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}]$

Step 1.3.1.6.3

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

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}]$

Step 1.3.1.6.4

Cancel the common factor of $2$ .

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Step 1.3.1.6.4.1

Cancel the common factor.

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}]$

Step 1.3.1.6.4.2

Rewrite the expression.

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2^{1}}]$

Step 1.3.1.6.5

Evaluate the exponent.

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2}]$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

$\frac{d}{d x} [\frac{2 \sqrt{2}}{2}]$

Step 1.3.2.2

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

$\frac{d}{d x} [\sqrt{2}]$

Step 1.4

Since $\sqrt{2}$ is constant with respect to $x$ , the derivative of $\sqrt{2}$ with respect to $x$ is $0$ .

$f' (x) = 0$

Step 2

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

$f'' (x) = 0$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $0$ .

$0$