Calculus Examples

$y = arcsec (3 x^{3})$

Step 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 $g (x) = 3 x^{3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{3}$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [3 x^{3}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $3 x^{3}$ .

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

Step 2

Differentiate.

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Step 2.1

Factor $3$ out of $3 x^{3}$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $3 (x^{3})$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.3.2

Multiply $3$ by $2$ .

$\frac{1}{3 x^{3} \sqrt{9 x^{6} - 1}} \frac{d}{d x} [3 x^{3}]$

Step 2.3

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

$\frac{1}{3 x^{3} \sqrt{9 x^{6} - 1}} (3 \frac{d}{d x} [x^{3}])$

Step 2.4

Simplify terms.

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Step 2.4.1

Combine $3$ and $\frac{1}{3 x^{3} \sqrt{9 x^{6} - 1}}$ .

$\frac{3}{3 x^{3} \sqrt{9 x^{6} - 1}} \frac{d}{d x} [x^{3}]$

Step 2.4.2

Cancel the common factor of $3$ .

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Step 2.4.2.1

Cancel the common factor.

$\frac{3}{3 x^{3} \sqrt{9 x^{6} - 1}} \frac{d}{d x} [x^{3}]$

Step 2.4.2.2

Rewrite the expression.

$\frac{1}{x^{3} \sqrt{9 x^{6} - 1}} \frac{d}{d x} [x^{3}]$

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{x^{3} \sqrt{9 x^{6} - 1}} (3 x^{2})$

Step 2.6

Simplify terms.

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Step 2.6.1

Combine $3$ and $\frac{1}{x^{3} \sqrt{9 x^{6} - 1}}$ .

$\frac{3}{x^{3} \sqrt{9 x^{6} - 1}} x^{2}$

Step 2.6.2

Combine $\frac{3}{x^{3} \sqrt{9 x^{6} - 1}}$ and $x^{2}$ .

$\frac{3 x^{2}}{x^{3} \sqrt{9 x^{6} - 1}}$

Step 2.6.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.6.3.1

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} \cdot 3}{x^{3} \sqrt{9 x^{6} - 1}}$

Step 2.6.3.2

Cancel the common factors.

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Step 2.6.3.2.1

Factor $x^{2}$ out of $x^{3} \sqrt{9 x^{6} - 1}$ .

$\frac{x^{2} \cdot 3}{x^{2} (x \sqrt{9 x^{6} - 1})}$

Step 2.6.3.2.2

Cancel the common factor.

$\frac{x^{2} \cdot 3}{x^{2} (x \sqrt{9 x^{6} - 1})}$

Step 2.6.3.2.3

Rewrite the expression.

$\frac{3}{x \sqrt{9 x^{6} - 1}}$