Calculus Examples

$f (z) = \sqrt{\sin (z)}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

$\frac{1}{2 {\sin (z)}^{\frac{1}{2}}} \cos (z)$

Step 9

Combine $\frac{1}{2 {\sin (z)}^{\frac{1}{2}}}$ and $\cos (z)$ .

$\frac{\cos (z)}{2 {\sin (z)}^{\frac{1}{2}}}$

Step 10

Simplify.

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Step 10.1

Separate fractions.

$\frac{\cos (z)}{2} \cdot \frac{1}{{\sin (z)}^{\frac{1}{2}}}$

Step 10.2

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

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

Step 10.3

Combine $\frac{\cos (z)}{2}$ and ${\csc (z)}^{\frac{1}{2}}$ .

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

Step 10.4

Reorder factors in $\frac{\cos (z) {\csc (z)}^{\frac{1}{2}}}{2}$ .

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