Calculus Examples

$y = arcsec (3 x) + arccsc (3 x)$

Step 1

By the Sum Rule, the derivative of $arcsec (3 x) + arccsc (3 x)$ with respect to $x$ is $\frac{d}{d x} [arcsec (3 x)] + \frac{d}{d x} [arccsc (3 x)]$ .

$\frac{d}{d x} [arcsec (3 x)] + \frac{d}{d x} [arccsc (3 x)]$

Step 2

Evaluate $\frac{d}{d x} [arcsec (3 x)]$ .

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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 $f (x) = arcsec (x)$ and $g (x) = 3 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$\frac{d}{d u_{1}} [arcsec (u_{1})] \frac{d}{d x} [3 x] + \frac{d}{d x} [arccsc (3 x)]$

Step 2.1.2

The derivative of $arcsec (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \sqrt{{u_{1}}^{2} - 1}}$ .

$\frac{1}{u_{1} \sqrt{{u_{1}}^{2} - 1}} \frac{d}{d x} [3 x] + \frac{d}{d x} [arccsc (3 x)]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$\frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} \frac{d}{d x} [3 x] + \frac{d}{d x} [arccsc (3 x)]$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} (3 \frac{d}{d x} [x]) + \frac{d}{d x} [arccsc (3 x)]$

Step 2.3

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

$\frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} (3 \cdot 1) + \frac{d}{d x} [arccsc (3 x)]$

Step 2.4

Factor $3$ out of $3 x$ .

$\frac{1}{3 x \sqrt{{(3 (x))}^{2} - 1}} (3 \cdot 1) + \frac{d}{d x} [arccsc (3 x)]$

Step 2.5

Apply the product rule to $3 (x)$ .

$\frac{1}{3 x \sqrt{3^{2} x^{2} - 1}} (3 \cdot 1) + \frac{d}{d x} [arccsc (3 x)]$

Step 2.6

Raise $3$ to the power of $2$ .

$\frac{1}{3 x \sqrt{9 x^{2} - 1}} (3 \cdot 1) + \frac{d}{d x} [arccsc (3 x)]$

Step 2.7

Multiply $3$ by $1$ .

$\frac{1}{3 x \sqrt{9 x^{2} - 1}} \cdot 3 + \frac{d}{d x} [arccsc (3 x)]$

Step 2.8

Combine $\frac{1}{3 x \sqrt{9 x^{2} - 1}}$ and $3$ .

$\frac{3}{3 x \sqrt{9 x^{2} - 1}} + \frac{d}{d x} [arccsc (3 x)]$

Step 2.9

Cancel the common factor of $3$ .

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Step 2.9.1

Cancel the common factor.

$\frac{3}{3 x \sqrt{9 x^{2} - 1}} + \frac{d}{d x} [arccsc (3 x)]$

Step 2.9.2

Rewrite the expression.

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{d}{d x} [arccsc (3 x)]$

Step 3

Evaluate $\frac{d}{d x} [arccsc (3 x)]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arccsc (x)$ and $g (x) = 3 x$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{d}{d u_{2}} [arccsc (u_{2})] \frac{d}{d x} [3 x]$

Step 3.1.2

The derivative of $arccsc (u_{2})$ with respect to $u_{2}$ is $- \frac{1}{u_{2} \sqrt{{u_{2}}^{2} - 1}}$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{u_{2} \sqrt{{u_{2}}^{2} - 1}} \frac{d}{d x} [3 x]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} \frac{d}{d x} [3 x]$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} (3 \frac{d}{d x} [x])$

Step 3.3

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

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{{(3 x)}^{2} - 1}} (3 \cdot 1)$

Step 3.4

Factor $3$ out of $3 x$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{{(3 (x))}^{2} - 1}} (3 \cdot 1)$

Step 3.5

Apply the product rule to $3 (x)$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{3^{2} x^{2} - 1}} (3 \cdot 1)$

Step 3.6

Raise $3$ to the power of $2$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{9 x^{2} - 1}} (3 \cdot 1)$

Step 3.7

Multiply $3$ by $1$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{3 x \sqrt{9 x^{2} - 1}} \cdot 3$

Step 3.8

Multiply $3$ by $- 1$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} - 3 \frac{1}{3 x \sqrt{9 x^{2} - 1}}$

Step 3.9

Combine $- 3$ and $\frac{1}{3 x \sqrt{9 x^{2} - 1}}$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{- 3}{3 x \sqrt{9 x^{2} - 1}}$

Step 3.10

Cancel the common factor of $- 3$ and $3$ .

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Step 3.10.1

Factor $3$ out of $- 3$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{3 \cdot - 1}{3 x \sqrt{9 x^{2} - 1}}$

Step 3.10.2

Cancel the common factors.

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Step 3.10.2.1

Factor $3$ out of $3 x \sqrt{9 x^{2} - 1}$ .

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{3 \cdot - 1}{3 (x \sqrt{9 x^{2} - 1})}$

Step 3.10.2.2

Cancel the common factor.

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{3 \cdot - 1}{3 (x \sqrt{9 x^{2} - 1})}$

Step 3.10.2.3

Rewrite the expression.

$\frac{1}{x \sqrt{9 x^{2} - 1}} + \frac{- 1}{x \sqrt{9 x^{2} - 1}}$

Step 3.11

Move the negative in front of the fraction.

$\frac{1}{x \sqrt{9 x^{2} - 1}} - \frac{1}{x \sqrt{9 x^{2} - 1}}$

Step 4

Subtract $\frac{1}{x \sqrt{9 x^{2} - 1}}$ from $\frac{1}{x \sqrt{9 x^{2} - 1}}$ .

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