Calculus Examples

$y = 5 \csc (x) \sec (x)$

Step 1

Find the first derivative.

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Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \csc (x) \sec (x)$ with respect to $x$ is $5 \frac{d}{d x} [\csc (x) \sec (x)]$ .

$5 \frac{d}{d x} [\csc (x) \sec (x)]$

Step 1.2

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 $f (x) = \csc (x)$ and $g (x) = \sec (x)$ .

$5 (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 1.3

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

$5 (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 1.4

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$5 (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$5 (\csc (x) \sec (x) \tan (x)) + 5 (\sec (x) (- \csc (x) \cot (x)))$

Step 1.5.2

Multiply $- 1$ by $5$ .

$5 \csc (x) \sec (x) \tan (x) - 5 \sec (x) \csc (x) \cot (x)$

Step 1.5.3

Reorder terms.

$f' (x) = 5 \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x)$ with respect to $x$ is $\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$ .

$\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \csc (x) \sec (x) \tan (x)$ with respect to $x$ is $5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$ .

$5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)] + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

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 $f (x) = \csc (x) \sec (x)$ and $g (x) = \tan (x)$ .

$5 (\csc (x) \sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.3

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.4

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 $f (x) = \csc (x)$ and $g (x) = \sec (x)$ .

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.5

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

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.6

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.7

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

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Step 2.2.7.1

Move $\sec^{2} (x)$ .

$5 (\csc (x) (\sec^{2} (x) \sec (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.7.2

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

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Step 2.2.7.2.1

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

$5 (\csc (x) (\sec^{2} (x) \sec^{1} (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.7.2.2

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

$5 (\csc (x) {\sec (x)}^{2 + 1} + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.2.7.3

Add $2$ and $1$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) + \frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$ .

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Step 2.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \cot (x) \csc (x) \sec (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)]$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)]$

Step 2.3.2

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 $f (x) = \cot (x) \csc (x)$ and $g (x) = \sec (x)$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)])$

Step 2.3.3

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)])$

Step 2.3.4

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 $f (x) = \cot (x)$ and $g (x) = \csc (x)$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot (x)]))$

Step 2.3.5

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)]))$

Step 2.3.6

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

Step 2.3.7

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) (- \csc^{2} (x))))$

Step 2.3.8

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) (- \csc^{2} (x))))$

Step 2.3.9

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) (- \csc^{2} (x))))$

Step 2.3.10

Add $1$ and $1$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x))))$

Step 2.3.11

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x))))$

Step 2.3.12

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

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2}))$

Step 2.3.13

Add $1$ and $2$ .

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Step 2.4.2

Apply the distributive property.

$5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x)))$

Step 2.4.3

Apply the distributive property.

Step 2.4.4

Apply the distributive property.

$5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5

Combine terms.

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Step 2.4.5.1

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

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.2

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

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan^{1} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.3

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

$5 \csc (x) \sec^{3} (x) + 5 ({\tan (x)}^{1 + 1} (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.4

Add $1$ and $1$ .

$5 \csc (x) \sec^{3} (x) + 5 (\tan^{2} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x)))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.5

Multiply $- 1$ by $5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 (\tan (x) \sec (x) (\csc (x) \cot (x))) - 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.6

Multiply $- 1$ by $- 5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \tan (x) \sec (x) \csc (x) \cot (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 (\sec (x) (\csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x)))$

Step 2.4.5.7

Multiply $- 1$ by $- 5$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \tan (x) \sec (x) \csc (x) \cot (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 (\sec (x) (\csc^{3} (x)))$

Step 2.4.5.8

Reorder the factors of $- 5 \tan (x) \sec (x) \csc (x) \cot (x)$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x)$

Step 2.4.5.9

Subtract $5 \cot (x) \csc (x) \sec (x) \tan (x)$ from $- 5 \cot (x) \csc (x) \sec (x) \tan (x)$ .

$5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x)$

Step 2.4.6

Reorder terms.

$f'' (x) = 5 \sec^{3} (x) \csc (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \cot^{2} (x) \csc (x) \sec (x) + 5 \csc^{3} (x) \sec (x)$