Calculus Examples

$- 2 \sqrt{\sin (x)}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (x)}$ as ${\sin (x)}^{\frac{1}{2}}$ .

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

Step 1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 {\sin (x)}^{\frac{1}{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [{\sin (x)}^{\frac{1}{2}}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \sin (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 (\frac{1}{2} {\sin (x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\sin (x)])$

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 5

Combine the numerators over the common denominator.

$- 2 (\frac{1}{2} {\sin (x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\sin (x)])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Combine $\frac{1}{2}$ and ${\sin (x)}^{- \frac{1}{2}}$ .

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

Step 9

Move ${\sin (x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

Combine $\frac{1}{2 {\sin (x)}^{\frac{1}{2}}}$ and $- 2$ .

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

Step 11

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2 {\sin (x)}^{\frac{1}{2}}} \frac{d}{d x} [\sin (x)]$

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {\sin (x)}^{\frac{1}{2}}$ .

$\frac{2 \cdot - 1}{2 ({\sin (x)}^{\frac{1}{2}})} \frac{d}{d x} [\sin (x)]$

Step 12.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 {\sin (x)}^{\frac{1}{2}}} \frac{d}{d x} [\sin (x)]$

Step 12.3

Rewrite the expression.

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

Step 13

Move the negative in front of the fraction.

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

Step 14

Convert from $\frac{1}{{\sin (x)}^{\frac{1}{2}}}$ to ${\csc (x)}^{\frac{1}{2}}$ .

$- {\csc (x)}^{\frac{1}{2}} \frac{d}{d x} [\sin (x)]$

Step 15

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$- {\csc (x)}^{\frac{1}{2}} \cos (x)$