Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $1$ and $2$ .

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

Step 3

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

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

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = \sin (x)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $3$ .

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Combine terms.

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Step 13.2.1

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

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

Step 13.2.2

Multiply $3$ by $2$ .

$2 x^{\frac{3}{2}} \cos (x) + \frac{6 (\sin (x) (x^{\frac{1}{2}}))}{2}$

Step 13.2.3

Factor $2$ out of $6 \sin (x) x^{\frac{1}{2}}$ .

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

Step 13.2.4

Cancel the common factors.

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Step 13.2.4.1

Factor $2$ out of $2$ .

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

Step 13.2.4.2

Cancel the common factor.

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

Step 13.2.4.3

Rewrite the expression.

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

Step 13.2.4.4

Divide $3 \sin (x) x^{\frac{1}{2}}$ by $1$ .

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

Step 13.3

Reorder terms.

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