Calculus Examples

$y = (\sec (x) + \tan (x)) (\sec (x) - \tan (x))$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (x)]$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify each term.

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Step 9.1.1

Expand $(\sec (x) + \tan (x)) (\sec (x) \tan (x) - \sec^{2} (x))$ using the FOIL Method.

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Step 9.1.1.1

Apply the distributive property.

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

Step 9.1.1.2

Apply the distributive property.

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

Step 9.1.1.3

Apply the distributive property.

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

Step 9.1.2

Simplify and combine like terms.

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Step 9.1.2.1

Simplify each term.

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Step 9.1.2.1.1

Multiply $\sec (x) (\sec (x) \tan (x))$ .

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Step 9.1.2.1.1.1

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

$\sec^{1} (x) \sec (x) \tan (x) + \sec (x) (- \sec^{2} (x)) + \tan (x) (\sec (x) \tan (x)) + \tan (x) (- \sec^{2} (x)) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.1.1.2

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

$\sec^{1} (x) \sec^{1} (x) \tan (x) + \sec (x) (- \sec^{2} (x)) + \tan (x) (\sec (x) \tan (x)) + \tan (x) (- \sec^{2} (x)) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.1.1.3

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

${\sec (x)}^{1 + 1} \tan (x) + \sec (x) (- \sec^{2} (x)) + \tan (x) (\sec (x) \tan (x)) + \tan (x) (- \sec^{2} (x)) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.1.1.4

Add $1$ and $1$ .

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

Step 9.1.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 9.1.2.1.3

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

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Step 9.1.2.1.3.1

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

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

Step 9.1.2.1.3.2

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

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Step 9.1.2.1.3.2.1

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

$\sec^{2} (x) \tan (x) - (\sec^{2} (x) \sec^{1} (x)) + \tan (x) (\sec (x) \tan (x)) + \tan (x) (- \sec^{2} (x)) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.1.3.2.2

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

$\sec^{2} (x) \tan (x) - {\sec (x)}^{2 + 1} + \tan (x) (\sec (x) \tan (x)) + \tan (x) (- \sec^{2} (x)) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.1.3.3

Add $2$ and $1$ .

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

Step 9.1.2.1.4

Multiply $\tan (x) (\sec (x) \tan (x))$ .

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Step 9.1.2.1.4.1

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

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

Step 9.1.2.1.4.2

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

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

Step 9.1.2.1.4.3

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

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

Step 9.1.2.1.4.4

Add $1$ and $1$ .

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

Step 9.1.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 9.1.2.2

Reorder the factors of $- \tan (x) \sec^{2} (x)$ .

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

Step 9.1.2.3

Subtract $\sec^{2} (x) \tan (x)$ from $\sec^{2} (x) \tan (x)$ .

$0 - \sec^{3} (x) + \tan^{2} (x) \sec (x) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.2.4

Subtract $\sec^{3} (x)$ from $0$ .

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

Step 9.1.3

Factor $\sec (x)$ out of $- \sec^{3} (x) + \tan^{2} (x) \sec (x)$ .

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Step 9.1.3.1

Factor $\sec (x)$ out of $- \sec^{3} (x)$ .

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

Step 9.1.3.2

Factor $\sec (x)$ out of $\tan^{2} (x) \sec (x)$ .

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

Step 9.1.3.3

Factor $\sec (x)$ out of $\sec (x) (- \sec^{2} (x)) + \sec (x) \tan^{2} (x)$ .

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

Step 9.1.4

Factor $- 1$ out of $- \sec^{2} (x)$ .

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

Step 9.1.5

Factor $- 1$ out of $\tan^{2} (x)$ .

$\sec (x) (- (\sec^{2} (x)) - 1 (- \tan^{2} (x))) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.6

Factor $- 1$ out of $- (\sec^{2} (x)) - 1 (- \tan^{2} (x))$ .

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

Step 9.1.7

Apply pythagorean identity.

$\sec (x) (- 1 \cdot 1) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.8

Multiply $- 1$ by $1$ .

$\sec (x) \cdot - 1 + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.9

Move $- 1$ to the left of $\sec (x)$ .

$- 1 \cdot \sec (x) + (\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$

Step 9.1.10

Rewrite $- 1 \sec (x)$ as $- \sec (x)$ .

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

Step 9.1.11

Expand $(\sec (x) - \tan (x)) (\sec (x) \tan (x) + \sec^{2} (x))$ using the FOIL Method.

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Step 9.1.11.1

Apply the distributive property.

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

Step 9.1.11.2

Apply the distributive property.

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

Step 9.1.11.3

Apply the distributive property.

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

Step 9.1.12

Simplify and combine like terms.

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Step 9.1.12.1

Simplify each term.

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Step 9.1.12.1.1

Multiply $\sec (x) (\sec (x) \tan (x))$ .

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Step 9.1.12.1.1.1

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

$- \sec (x) + \sec^{1} (x) \sec (x) \tan (x) + \sec (x) \sec^{2} (x) - \tan (x) (\sec (x) \tan (x)) - \tan (x) \sec^{2} (x)$

Step 9.1.12.1.1.2

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

$- \sec (x) + \sec^{1} (x) \sec^{1} (x) \tan (x) + \sec (x) \sec^{2} (x) - \tan (x) (\sec (x) \tan (x)) - \tan (x) \sec^{2} (x)$

Step 9.1.12.1.1.3

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

$- \sec (x) + {\sec (x)}^{1 + 1} \tan (x) + \sec (x) \sec^{2} (x) - \tan (x) (\sec (x) \tan (x)) - \tan (x) \sec^{2} (x)$

Step 9.1.12.1.1.4

Add $1$ and $1$ .

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

Step 9.1.12.1.2

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

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Step 9.1.12.1.2.1

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

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Step 9.1.12.1.2.1.1

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

$- \sec (x) + \sec^{2} (x) \tan (x) + \sec^{1} (x) \sec^{2} (x) - \tan (x) (\sec (x) \tan (x)) - \tan (x) \sec^{2} (x)$

Step 9.1.12.1.2.1.2

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

$- \sec (x) + \sec^{2} (x) \tan (x) + {\sec (x)}^{1 + 2} - \tan (x) (\sec (x) \tan (x)) - \tan (x) \sec^{2} (x)$

Step 9.1.12.1.2.2

Add $1$ and $2$ .

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

Step 9.1.12.1.3

Multiply $- \tan (x) (\sec (x) \tan (x))$ .

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Step 9.1.12.1.3.1

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

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

Step 9.1.12.1.3.2

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

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

Step 9.1.12.1.3.3

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

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

Step 9.1.12.1.3.4

Add $1$ and $1$ .

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

Step 9.1.12.2

Reorder the factors of $- \tan (x) \sec^{2} (x)$ .

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

Step 9.1.12.3

Subtract $\sec^{2} (x) \tan (x)$ from $\sec^{2} (x) \tan (x)$ .

$- \sec (x) + 0 + \sec^{3} (x) - \tan^{2} (x) \sec (x)$

Step 9.1.12.4

Add $0$ and $\sec^{3} (x)$ .

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

Step 9.1.13

Factor $\sec (x)$ out of $\sec^{3} (x) - \tan^{2} (x) \sec (x)$ .

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Step 9.1.13.1

Factor $\sec (x)$ out of $\sec^{3} (x)$ .

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

Step 9.1.13.2

Factor $\sec (x)$ out of $- \tan^{2} (x) \sec (x)$ .

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

Step 9.1.13.3

Factor $\sec (x)$ out of $\sec (x) \sec^{2} (x) + \sec (x) (- \tan^{2} (x))$ .

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

Step 9.1.14

Apply pythagorean identity.

$- \sec (x) + \sec (x) \cdot 1$

Step 9.1.15

Multiply $\sec (x)$ by $1$ .

$- \sec (x) + \sec (x)$

Step 9.2

Add $- \sec (x)$ and $\sec (x)$ .

$0$