Calculus Examples

$\sqrt{3 \cos^{3} (x)}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{3 \cos^{3} (x)}]$ as $\frac{d}{d x} [\cos (x) \sqrt{3 \cos (x)}]$ .

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Step 1.1

Rewrite $3 \cos^{3} (x)$ as $\cos^{2} (x) \cdot (3 \cos (x))$ .

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Step 1.1.1

Factor out $\cos^{2} (x)$ .

$\frac{d}{d x} [\sqrt{3 (\cos^{2} (x) \cos (x))}]$

Step 1.1.2

Reorder $3$ and $\cos^{2} (x)$ .

$\frac{d}{d x} [\sqrt{\cos^{2} (x) \cdot 3 \cos (x)}]$

Step 1.1.3

Add parentheses.

$\frac{d}{d x} [\sqrt{\cos^{2} (x) \cdot (3 \cos (x))}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\cos (x) \sqrt{3 \cos (x)}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 \cos (x)}$ as ${(3 \cos (x))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\cos (x) {(3 \cos (x))}^{\frac{1}{2}}]$

Step 3

Factor $3$ out of $3 \cos (x)$ .

$\frac{d}{d x} [\cos (x) {(3 (\cos (x)))}^{\frac{1}{2}}]$

Step 4

Apply the product rule to $3 (\cos (x))$ .

$\frac{d}{d x} [\cos (x) (3^{\frac{1}{2}} {\cos (x)}^{\frac{1}{2}})]$

Step 5

Raise $\cos (x)$ to the power of $1$ .

$\frac{d}{d x} [3^{\frac{1}{2}} (\cos^{1} (x) {\cos (x)}^{\frac{1}{2}})]$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [3^{\frac{1}{2}} {\cos (x)}^{1 + \frac{1}{2}}]$

Step 7

Differentiate using the Constant Multiple Rule.

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Step 7.1

Write $1$ as a fraction with a common denominator.

$\frac{d}{d x} [3^{\frac{1}{2}} {\cos (x)}^{\frac{2}{2} + \frac{1}{2}}]$

Step 7.2

Simplify the expression.

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Step 7.2.1

Combine the numerators over the common denominator.

$\frac{d}{d x} [3^{\frac{1}{2}} {\cos (x)}^{\frac{2 + 1}{2}}]$

Step 7.2.2

Add $2$ and $1$ .

$\frac{d}{d x} [3^{\frac{1}{2}} {\cos (x)}^{\frac{3}{2}}]$

Step 7.3

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} {\cos (x)}^{\frac{3}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [{\cos (x)}^{\frac{3}{2}}]$ .

$3^{\frac{1}{2}} \frac{d}{d x} [{\cos (x)}^{\frac{3}{2}}]$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = \cos (x)$ .

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Step 8.1

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

$3^{\frac{1}{2}} (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [\cos (x)])$

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$3^{\frac{1}{2}} (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [\cos (x)])$

Step 8.3

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

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{3}{2} - 1} \frac{d}{d x} [\cos (x)])$

Step 9

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

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\cos (x)])$

Step 10

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

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\cos (x)])$

Step 11

Combine the numerators over the common denominator.

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [\cos (x)])$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $2$ .

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{3 - 2}{2}} \frac{d}{d x} [\cos (x)])$

Step 12.2

Subtract $2$ from $3$ .

$3^{\frac{1}{2}} (\frac{3}{2} {\cos (x)}^{\frac{1}{2}} \frac{d}{d x} [\cos (x)])$

Step 13

Combine $\frac{3}{2}$ and ${\cos (x)}^{\frac{1}{2}}$ .

$3^{\frac{1}{2}} (\frac{3 {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)])$

Step 14

Combine $\frac{3 {\cos (x)}^{\frac{1}{2}}}{2}$ and $3^{\frac{1}{2}}$ .

$\frac{3 {\cos (x)}^{\frac{1}{2}} \cdot 3^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15

Multiply $3$ by $3^{\frac{1}{2}}$ by adding the exponents.

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Step 15.1

Move $3^{\frac{1}{2}}$ .

$\frac{3^{\frac{1}{2}} \cdot 3 {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15.2

Multiply $3^{\frac{1}{2}}$ by $3$ .

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Step 15.2.1

Raise $3$ to the power of $1$ .

$\frac{3^{\frac{1}{2}} \cdot 3^{1} {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{\frac{1}{2} + 1} {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15.3

Write $1$ as a fraction with a common denominator.

$\frac{3^{\frac{1}{2} + \frac{2}{2}} {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15.4

Combine the numerators over the common denominator.

$\frac{3^{\frac{1 + 2}{2}} {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 15.5

Add $1$ and $2$ .

$\frac{3^{\frac{3}{2}} {\cos (x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (x)]$

Step 16

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

$\frac{3^{\frac{3}{2}} {\cos (x)}^{\frac{1}{2}}}{2} (- \sin (x))$

Step 17

Combine $\frac{3^{\frac{3}{2}} {\cos (x)}^{\frac{1}{2}}}{2}$ and $\sin (x)$ .

$- \frac{3^{\frac{3}{2}} {\cos (x)}^{\frac{1}{2}} \sin (x)}{2}$