Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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