Calculus Examples

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

Step 1

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

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

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 $g (x) = \sin (x^{2})$ .

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To apply the Chain Rule, set $u_{1}$ as $\sin (x^{2})$ .

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

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

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

Replace all occurrences of $u_{1}$ with $\sin (x^{2})$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

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Move the negative in front of the fraction.

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

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

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

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

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

Step 8

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

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To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

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

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Replace all occurrences of $u_{2}$ with $x^{2}$ .

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

Step 9

Differentiate using the Power Rule.

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

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

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

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

Simplify terms.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Reorder factors in $\cos (x^{2}) x$ .

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