Calculus Examples

$f (x) = 8 \sec (x) + 4 \tan (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $8 \sec (x) + 4 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [8 \sec (x)] + \frac{d}{d x} [4 \tan (x)]$ .

$\frac{d}{d x} [8 \sec (x)] + \frac{d}{d x} [4 \tan (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [8 \sec (x)]$ .

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Step 1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sec (x)$ with respect to $x$ is $8 \frac{d}{d x} [\sec (x)]$ .

$8 \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [4 \tan (x)]$

Step 1.2.2

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

$8 \sec (x) \tan (x) + \frac{d}{d x} [4 \tan (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [4 \tan (x)]$ .

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \tan (x)$ with respect to $x$ is $4 \frac{d}{d x} [\tan (x)]$ .

$8 \sec (x) \tan (x) + 4 \frac{d}{d x} [\tan (x)]$

Step 1.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$8 \sec (x) \tan (x) + 4 \sec^{2} (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $8 \sec (x) \tan (x) + 4 \sec^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [8 \sec (x) \tan (x)] + \frac{d}{d x} [4 \sec^{2} (x)]$ .

$f'' (x) = \frac{d}{d x} (8 \sec (x) \tan (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2

Evaluate $\frac{d}{d x} [8 \sec (x) \tan (x)]$ .

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Step 2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sec (x) \tan (x)$ with respect to $x$ is $8 \frac{d}{d x} [\sec (x) \tan (x)]$ .

$f'' (x) = 8 \frac{d}{d x} (\sec (x) \tan (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \tan (x)$ .

$f'' (x) = 8 (\sec (x) \frac{d}{d x} (\tan (x)) + \tan (x) \frac{d}{d x} (\sec (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$f'' (x) = 8 (\sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} (\sec (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.4

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

$f'' (x) = 8 (\sec (x) \sec^{2} (x) + \tan (x) (\sec (x) \tan (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.5

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 2.2.5.1

Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Step 2.2.5.1.1

Raise $\sec (x)$ to the power of $1$ .

$f'' (x) = 8 (\sec (x) \sec^{2} (x) + \tan (x) (\sec (x) \tan (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.5.1.2

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

$f'' (x) = 8 ({\sec (x)}^{1 + 2} + \tan (x) (\sec (x) \tan (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.5.2

Add $1$ and $2$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan (x) (\sec (x) \tan (x))) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.6

Raise $\tan (x)$ to the power of $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan (x) \tan (x) \sec (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.7

Raise $\tan (x)$ to the power of $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan (x) \tan (x) \sec (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.8

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

$f'' (x) = 8 (\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.2.9

Add $1$ and $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + \frac{d}{d x} (4 \sec^{2} (x))$

Step 2.3

Evaluate $\frac{d}{d x} [4 \sec^{2} (x)]$ .

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Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \sec^{2} (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sec^{2} (x)]$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 \frac{d}{d x} (\sec^{2} (x))$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (x)$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (\frac{d}{d u} (u^{2}) \frac{d}{d x} (\sec (x)))$

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 u \frac{d}{d x} (\sec (x)))$

Step 2.3.2.3

Replace all occurrences of $u$ with $\sec (x)$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 \sec (x) \frac{d}{d x} (\sec (x)))$

Step 2.3.3

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

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 \sec (x) (\sec (x) \tan (x)))$

Step 2.3.4

Raise $\sec (x)$ to the power of $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 (\sec (x) \sec (x)) \tan (x))$

Step 2.3.5

Raise $\sec (x)$ to the power of $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 (\sec (x) \sec (x)) \tan (x))$

Step 2.3.6

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

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 {\sec (x)}^{1 + 1} \tan (x))$

Step 2.3.7

Add $1$ and $1$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 4 (2 \sec^{2} (x) \tan (x))$

Step 2.3.8

Multiply $2$ by $4$ .

$f'' (x) = 8 (\sec^{3} (x) + \tan^{2} (x) \sec (x)) + 8 \sec^{2} (x) \tan (x)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = 8 \sec^{3} (x) + 8 (\tan^{2} (x) \sec (x)) + 8 \sec^{2} (x) \tan (x)$

Step 2.4.2

Reorder terms.

$f'' (x) = 8 \tan^{2} (x) \sec (x) + 8 \sec^{2} (x) \tan (x) + 8 \sec^{3} (x)$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$8 \sec (x) \tan (x) + 4 \sec^{2} (x) = 0$

Step 4

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 5

Evaluate the second derivative at $x = \frac{7 π}{6} + 2 π n$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$8 \tan^{2} (\frac{7 π}{6} + 2 π n) \sec (\frac{7 π}{6} + 2 π n) + 8 \sec^{2} (\frac{7 π}{6} + 2 π n) \tan (\frac{7 π}{6} + 2 π n) + 8 \sec^{3} (\frac{7 π}{6} + 2 π n)$

Step 6

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 7