Calculus Examples

$f (x) = 8 \sin^{6} (\sqrt{x})$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [8 \sin^{6} (x^{\frac{1}{2}})]$

Step 1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 \sin^{6} (x^{\frac{1}{2}})$ with respect to $x$ is $8 \frac{d}{d x} [\sin^{6} (x^{\frac{1}{2}})]$ .

$8 \frac{d}{d x} [\sin^{6} (x^{\frac{1}{2}})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = \sin (x^{\frac{1}{2}})$ .

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Step 2.1

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

$8 (\frac{d}{d u_{1}} [{u_{1}}^{6}] \frac{d}{d x} [\sin (x^{\frac{1}{2}})])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 6$ .

$8 (6 {u_{1}}^{5} \frac{d}{d x} [\sin (x^{\frac{1}{2}})])$

Step 2.3

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

$8 (6 \sin^{5} (x^{\frac{1}{2}}) \frac{d}{d x} [\sin (x^{\frac{1}{2}})])$

Step 3

Multiply $6$ by $8$ .

$48 (\sin^{5} (x^{\frac{1}{2}}) \frac{d}{d x} [\sin (x^{\frac{1}{2}})])$

Step 4

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 4.1

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

$48 \sin^{5} (x^{\frac{1}{2}}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.2

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

$48 \sin^{5} (x^{\frac{1}{2}}) (\cos (u_{2}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 4.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{2}}$ .

$48 \sin^{5} (x^{\frac{1}{2}}) (\cos (x^{\frac{1}{2}}) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 5

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

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 6

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

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 7

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

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 8

Combine the numerators over the common denominator.

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 9.2

Subtract $2$ from $1$ .

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 10

Move the negative in front of the fraction.

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) (\frac{1}{2} x^{- \frac{1}{2}})$

Step 11

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

$48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}) \frac{x^{- \frac{1}{2}}}{2}$

Step 12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $48$ .

$\frac{x^{- \frac{1}{2}} \cdot 48}{2} \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})$

Step 13

Combine $\frac{x^{- \frac{1}{2}} \cdot 48}{2}$ and $\sin^{5} (x^{\frac{1}{2}})$ .

$\frac{x^{- \frac{1}{2}} \cdot 48 \sin^{5} (x^{\frac{1}{2}})}{2} \cos (x^{\frac{1}{2}})$

Step 14

Combine $\frac{x^{- \frac{1}{2}} \cdot 48 \sin^{5} (x^{\frac{1}{2}})}{2}$ and $\cos (x^{\frac{1}{2}})$ .

$\frac{x^{- \frac{1}{2}} \cdot 48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})}{2}$

Step 15

Simplify the expression.

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Step 15.1

Move $48$ to the left of $x^{- \frac{1}{2}}$ .

$\frac{48 \cdot x^{- \frac{1}{2}} \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})}{2}$

Step 15.2

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

$\frac{48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}$

Step 16

Factor $2$ out of $48 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})$ .

$\frac{2 (24 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}))}{2 x^{\frac{1}{2}}}$

Step 17

Cancel the common factors.

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Step 17.1

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

$\frac{2 (24 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}))}{2 (x^{\frac{1}{2}})}$

Step 17.2

Cancel the common factor.

$\frac{2 (24 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}}))}{2 x^{\frac{1}{2}}}$

Step 17.3

Rewrite the expression.

$\frac{24 \sin^{5} (x^{\frac{1}{2}}) \cos (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$