Calculus Examples

$9 \arcsin (6 x^{2} + 19 x - 9)$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 \arcsin (6 x^{2} + 19 x - 9)$ with respect to $x$ is $9 \frac{d}{d x} [\arcsin (6 x^{2} + 19 x - 9)]$ .

$9 \frac{d}{d x} [\arcsin (6 x^{2} + 19 x - 9)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = 6 x^{2} + 19 x - 9$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x^{2} + 19 x - 9$ .

$9 (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [6 x^{2} + 19 x - 9])$

Step 2.2

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

$9 (\frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [6 x^{2} + 19 x - 9])$

Step 2.3

Replace all occurrences of $u$ with $6 x^{2} + 19 x - 9$ .

$9 (\frac{1}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} \frac{d}{d x} [6 x^{2} + 19 x - 9])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}}$ and $9$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} \frac{d}{d x} [6 x^{2} + 19 x - 9]$

Step 3.2

By the Sum Rule, the derivative of $6 x^{2} + 19 x - 9$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [19 x] + \frac{d}{d x} [- 9]$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [19 x] + \frac{d}{d x} [- 9])$

Step 3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [19 x] + \frac{d}{d x} [- 9])$

Step 3.4

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

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (6 (2 x) + \frac{d}{d x} [19 x] + \frac{d}{d x} [- 9])$

Step 3.5

Multiply $2$ by $6$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + \frac{d}{d x} [19 x] + \frac{d}{d x} [- 9])$

Step 3.6

Since $19$ is constant with respect to $x$ , the derivative of $19 x$ with respect to $x$ is $19 \frac{d}{d x} [x]$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])$

Step 3.7

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

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19 \cdot 1 + \frac{d}{d x} [- 9])$

Step 3.8

Multiply $19$ by $1$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19 + \frac{d}{d x} [- 9])$

Step 3.9

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19 + 0)$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $12 x + 19$ and $0$ .

$\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19)$

Step 3.10.2

Reorder the factors of $\frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}} (12 x + 19)$ .

$(12 x + 19) \frac{9}{\sqrt{1 - {(6 x^{2} + 19 x - 9)}^{2}}}$