Calculus Examples

$y = \frac{\tan (x)}{\sqrt{x}}$

Step 1

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

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \tan (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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

Step 3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 6

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{1}{2} - 1})}{x}$

Step 7

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}{x}$

Step 8

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}{x}$

Step 9

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}{x}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{1 - 2}{2}})}{x}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{\frac{- 1}{2}})}{x}$

Step 11

Move the negative in front of the fraction.

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) (\frac{1}{2} x^{- \frac{1}{2}})}{x}$

Step 12

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \tan (x) \frac{x^{- \frac{1}{2}}}{2}}{x}$

Step 13

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \frac{x^{- \frac{1}{2}} \tan (x)}{2}}{x}$

Step 14

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

$\frac{x^{\frac{1}{2}} \sec^{2} (x) - \frac{\tan (x)}{2 x^{\frac{1}{2}}}}{x}$