Calculus Examples

$y = \tan (x - \frac{π}{6})$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{6}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x - \frac{π}{6}$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [x - \frac{π}{6}]$

Step 1.1.2

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

$\sec^{2} (u) \frac{d}{d x} [x - \frac{π}{6}]$

Step 1.1.3

Replace all occurrences of $u$ with $x - \frac{π}{6}$ .

$\sec^{2} (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]$ .

$\sec^{2} (x - \frac{π}{6}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}])$

Step 1.2.2

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

$\sec^{2} (x - \frac{π}{6}) (1 + \frac{d}{d x} [- \frac{π}{6}])$

Step 1.2.3

Since $- \frac{π}{6}$ is constant with respect to $x$ , the derivative of $- \frac{π}{6}$ with respect to $x$ is $0$ .

$\sec^{2} (x - \frac{π}{6}) (1 + 0)$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$\sec^{2} (x - \frac{π}{6}) \cdot 1$

Step 1.2.4.2

Multiply $\sec^{2} (x - \frac{π}{6})$ by $1$ .

$f' (x) = \sec^{2} (x - \frac{π}{6})$

Step 2

Find the second derivative.

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Step 2.1

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 $g (x) = \sec (x - \frac{π}{6})$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $\sec (x - \frac{π}{6})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sec (x - \frac{π}{6})]$

Step 2.1.2

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

$2 u_{1} \frac{d}{d x} [\sec (x - \frac{π}{6})]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $\sec (x - \frac{π}{6})$ .

$2 \sec (x - \frac{π}{6}) \frac{d}{d x} [\sec (x - \frac{π}{6})]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{6}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{2}$ as $x - \frac{π}{6}$ .

$2 \sec (x - \frac{π}{6}) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.2.2

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

$2 \sec (x - \frac{π}{6}) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.2.3

Replace all occurrences of $u_{2}$ with $x - \frac{π}{6}$ .

$2 \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.3

Raise $\sec (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{1} (x - \frac{π}{6}) \sec (x - \frac{π}{6})) (\tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.4

Raise $\sec (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{1} (x - \frac{π}{6}) \sec^{1} (x - \frac{π}{6})) (\tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.5

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

$2 {\sec (x - \frac{π}{6})}^{1 + 1} (\tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.6

Add $1$ and $1$ .

$2 \sec^{2} (x - \frac{π}{6}) (\tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 2.7

By the Sum Rule, the derivative of $x - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]$ .

$2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}])$

Step 2.8

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

$2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6}) (1 + \frac{d}{d x} [- \frac{π}{6}])$

Step 2.9

Since $- \frac{π}{6}$ is constant with respect to $x$ , the derivative of $- \frac{π}{6}$ with respect to $x$ is $0$ .

$2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6}) (1 + 0)$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $1$ and $0$ .

$2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6}) \cdot 1$

Step 2.10.2

Multiply $2$ by $1$ .

$f'' (x) = 2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6})$

Step 3

Find the third derivative.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6})$ with respect to $x$ is $2 \frac{d}{d x} [\sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6})]$ .

$2 \frac{d}{d x} [\sec^{2} (x - \frac{π}{6}) \tan (x - \frac{π}{6})]$

Step 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 $f (x) = \sec^{2} (x - \frac{π}{6})$ and $g (x) = \tan (x - \frac{π}{6})$ .

$2 (\sec^{2} (x - \frac{π}{6}) \frac{d}{d x} [\tan (x - \frac{π}{6})] + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{6}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $x - \frac{π}{6}$ .

$2 (\sec^{2} (x - \frac{π}{6}) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [x - \frac{π}{6}]) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.3.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$2 (\sec^{2} (x - \frac{π}{6}) (\sec^{2} (u_{1}) \frac{d}{d x} [x - \frac{π}{6}]) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $x - \frac{π}{6}$ .

$2 (\sec^{2} (x - \frac{π}{6}) (\sec^{2} (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.4

Multiply $\sec^{2} (x - \frac{π}{6})$ by $\sec^{2} (x - \frac{π}{6})$ by adding the exponents.

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Step 3.4.1

Move $\sec^{2} (x - \frac{π}{6})$ .

$2 (\sec^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}] + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.4.2

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

$2 ({\sec (x - \frac{π}{6})}^{2 + 2} \frac{d}{d x} [x - \frac{π}{6}] + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.4.3

Add $2$ and $2$ .

$2 (\sec^{4} (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}] + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.5

Differentiate.

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Step 3.5.1

By the Sum Rule, the derivative of $x - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]$ .

$2 (\sec^{4} (x - \frac{π}{6}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.5.2

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

$2 (\sec^{4} (x - \frac{π}{6}) (1 + \frac{d}{d x} [- \frac{π}{6}]) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.5.3

Since $- \frac{π}{6}$ is constant with respect to $x$ , the derivative of $- \frac{π}{6}$ with respect to $x$ is $0$ .

$2 (\sec^{4} (x - \frac{π}{6}) (1 + 0) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $0$ .

$2 (\sec^{4} (x - \frac{π}{6}) \cdot 1 + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.5.4.2

Multiply $\sec^{4} (x - \frac{π}{6})$ by $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + \tan (x - \frac{π}{6}) \frac{d}{d x} [\sec^{2} (x - \frac{π}{6})])$

Step 3.6

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 $g (x) = \sec (x - \frac{π}{6})$ .

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Step 3.6.1

To apply the Chain Rule, set $u_{2}$ as $\sec (x - \frac{π}{6})$ .

$2 (\sec^{4} (x - \frac{π}{6}) + \tan (x - \frac{π}{6}) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\sec (x - \frac{π}{6})]))$

Step 3.6.2

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

$2 (\sec^{4} (x - \frac{π}{6}) + \tan (x - \frac{π}{6}) (2 u_{2} \frac{d}{d x} [\sec (x - \frac{π}{6})]))$

Step 3.6.3

Replace all occurrences of $u_{2}$ with $\sec (x - \frac{π}{6})$ .

$2 (\sec^{4} (x - \frac{π}{6}) + \tan (x - \frac{π}{6}) (2 \sec (x - \frac{π}{6}) \frac{d}{d x} [\sec (x - \frac{π}{6})]))$

Step 3.7

Move $2$ to the left of $\tan (x - \frac{π}{6})$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \cdot \tan (x - \frac{π}{6}) \sec (x - \frac{π}{6}) \frac{d}{d x} [\sec (x - \frac{π}{6})])$

Step 3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{6}$ .

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Step 3.8.1

To apply the Chain Rule, set $u_{3}$ as $x - \frac{π}{6}$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan (x - \frac{π}{6}) \sec (x - \frac{π}{6}) (\frac{d}{d u_{3}} [\sec (u_{3})] \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.8.2

The derivative of $\sec (u_{3})$ with respect to $u_{3}$ is $\sec (u_{3}) \tan (u_{3})$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan (x - \frac{π}{6}) \sec (x - \frac{π}{6}) (\sec (u_{3}) \tan (u_{3}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.8.3

Replace all occurrences of $u_{3}$ with $x - \frac{π}{6}$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan (x - \frac{π}{6}) \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \tan (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.9

Raise $\tan (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 (\tan^{1} (x - \frac{π}{6}) \tan (x - \frac{π}{6})) \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.10

Raise $\tan (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 (\tan^{1} (x - \frac{π}{6}) \tan^{1} (x - \frac{π}{6})) \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.11

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

$2 (\sec^{4} (x - \frac{π}{6}) + 2 {\tan (x - \frac{π}{6})}^{1 + 1} \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.12

Add $1$ and $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec (x - \frac{π}{6}) (\sec (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}]))$

Step 3.13

Raise $\sec (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) (\sec^{1} (x - \frac{π}{6}) \sec (x - \frac{π}{6})) \frac{d}{d x} [x - \frac{π}{6}])$

Step 3.14

Raise $\sec (x - \frac{π}{6})$ to the power of $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) (\sec^{1} (x - \frac{π}{6}) \sec^{1} (x - \frac{π}{6})) \frac{d}{d x} [x - \frac{π}{6}])$

Step 3.15

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

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) {\sec (x - \frac{π}{6})}^{1 + 1} \frac{d}{d x} [x - \frac{π}{6}])$

Step 3.16

Add $1$ and $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) \frac{d}{d x} [x - \frac{π}{6}])$

Step 3.17

By the Sum Rule, the derivative of $x - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{6}]))$

Step 3.18

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

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) (1 + \frac{d}{d x} [- \frac{π}{6}]))$

Step 3.19

Since $- \frac{π}{6}$ is constant with respect to $x$ , the derivative of $- \frac{π}{6}$ with respect to $x$ is $0$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) (1 + 0))$

Step 3.20

Simplify the expression.

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Step 3.20.1

Add $1$ and $0$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}) \cdot 1)$

Step 3.20.2

Multiply $2$ by $1$ .

$2 (\sec^{4} (x - \frac{π}{6}) + 2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}))$

Step 3.21

Simplify.

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Step 3.21.1

Apply the distributive property.

$2 \sec^{4} (x - \frac{π}{6}) + 2 (2 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6}))$

Step 3.21.2

Multiply $2$ by $2$ .

$2 \sec^{4} (x - \frac{π}{6}) + 4 \tan^{2} (x - \frac{π}{6}) \sec^{2} (x - \frac{π}{6})$

Step 3.21.3

Reorder terms.

$f''' (x) = 4 \sec^{2} (x - \frac{π}{6}) \tan^{2} (x - \frac{π}{6}) + 2 \sec^{4} (x - \frac{π}{6})$