Calculus Examples

$y = \sec^{-1} (2 s^{3} + 4)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = \sec^{-1} (s)$ and $g (s) = 2 s^{3} + 4$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 s^{3} + 4$ .

$\frac{d}{d u} [\sec^{-1} (u)] \frac{d}{d s} [2 s^{3} + 4]$

Step 1.2

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

$\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d s} [2 s^{3} + 4]$

Step 1.3

Replace all occurrences of $u$ with $2 s^{3} + 4$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} \frac{d}{d s} [2 s^{3} + 4]$

Step 2

By the Sum Rule, the derivative of $2 s^{3} + 4$ with respect to $s$ is $\frac{d}{d s} [2 s^{3}] + \frac{d}{d s} [4]$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (\frac{d}{d s} [2 s^{3}] + \frac{d}{d s} [4])$

Step 3

Evaluate $\frac{d}{d s} [2 s^{3}]$ .

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Step 3.1

Since $2$ is constant with respect to $s$ , the derivative of $2 s^{3}$ with respect to $s$ is $2 \frac{d}{d s} [s^{3}]$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (2 \frac{d}{d s} [s^{3}] + \frac{d}{d s} [4])$

Step 3.2

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

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (2 (3 s^{2}) + \frac{d}{d s} [4])$

Step 3.3

Multiply $3$ by $2$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (6 s^{2} + \frac{d}{d s} [4])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $4$ is constant with respect to $s$ , the derivative of $4$ with respect to $s$ is $0$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (6 s^{2} + 0)$

Step 4.2

Add $6 s^{2}$ and $0$ .

$\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} (6 s^{2})$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $6$ and $\frac{1}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}}$ .

$\frac{6}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}} s^{2}$

Step 5.1.2

Combine $\frac{6}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}}$ and $s^{2}$ .

$\frac{6 s^{2}}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}}$

Step 5.1.3

Cancel the common factor of $6$ and $2 s^{3} + 4$ .

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Step 5.1.3.1

Factor $2$ out of $6 s^{2}$ .

$\frac{2 (3 s^{2})}{(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}}$

Step 5.1.3.2

Cancel the common factors.

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Step 5.1.3.2.1

Factor $2$ out of $(2 s^{3} + 4) \sqrt{{(2 s^{3} + 4)}^{2} - 1}$ .

$\frac{2 (3 s^{2})}{2 ((s^{3} + 2) \sqrt{{(2 s^{3} + 4)}^{2} - 1})}$

Step 5.1.3.2.2

Cancel the common factor.

$\frac{2 (3 s^{2})}{2 ((s^{3} + 2) \sqrt{{(2 s^{3} + 4)}^{2} - 1})}$

Step 5.1.3.2.3

Rewrite the expression.

$\frac{3 s^{2}}{(s^{3} + 2) \sqrt{{(2 s^{3} + 4)}^{2} - 1}}$

Step 5.2

Reorder terms.

$\frac{3 s^{2}}{\sqrt{{(2 s^{3} + 4)}^{2} - 1} (s^{3} + 2)}$