Calculus Examples

$7 arcsec (7 x)$

Step 1

Since $7$ is constant with respect to $x$ , the derivative of $7 arcsec (7 x)$ with respect to $x$ is $7 \frac{d}{d x} [arcsec (7 x)]$ .

$7 \frac{d}{d x} [arcsec (7 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arcsec (x)$ and $g (x) = 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $7 x$ .

$7 (\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [7 x])$

Step 2.2

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

$7 (\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d x} [7 x])$

Step 2.3

Replace all occurrences of $u$ with $7 x$ .

$7 (\frac{1}{7 x \sqrt{{(7 x)}^{2} - 1}} \frac{d}{d x} [7 x])$

Step 3

Differentiate.

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Step 3.1

Factor $7$ out of $7 x$ .

$7 (\frac{1}{7 x \sqrt{{(7 (x))}^{2} - 1}} \frac{d}{d x} [7 x])$

Step 3.2

Simplify terms.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Apply the product rule to $7 (x)$ .

$7 (\frac{1}{7 x \sqrt{7^{2} x^{2} - 1}} \frac{d}{d x} [7 x])$

Step 3.2.1.2

Raise $7$ to the power of $2$ .

$7 (\frac{1}{7 x \sqrt{49 x^{2} - 1}} \frac{d}{d x} [7 x])$

Step 3.2.2

Combine $\frac{1}{7 x \sqrt{49 x^{2} - 1}}$ and $7$ .

$\frac{7}{7 x \sqrt{49 x^{2} - 1}} \frac{d}{d x} [7 x]$

Step 3.2.3

Cancel the common factor of $7$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{7}{7 x \sqrt{49 x^{2} - 1}} \frac{d}{d x} [7 x]$

Step 3.2.3.2

Rewrite the expression.

$\frac{1}{x \sqrt{49 x^{2} - 1}} \frac{d}{d x} [7 x]$

Step 3.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{1}{x \sqrt{49 x^{2} - 1}} (7 \frac{d}{d x} [x])$

Step 3.4

Combine $7$ and $\frac{1}{x \sqrt{49 x^{2} - 1}}$ .

$\frac{7}{x \sqrt{49 x^{2} - 1}} \frac{d}{d x} [x]$

Step 3.5

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

$\frac{7}{x \sqrt{49 x^{2} - 1}} \cdot 1$

Step 3.6

Multiply $\frac{7}{x \sqrt{49 x^{2} - 1}}$ by $1$ .

$\frac{7}{x \sqrt{49 x^{2} - 1}}$