Calculus Examples

$\frac{\sqrt{x}}{\cos (x)}$

Step 1

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Combine.

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

Step 14

Apply the distributive property.

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

Step 15

Cancel the common factor of $2 x^{\frac{1}{2}}$ .

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Step 15.1

Cancel the common factor.

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

Step 15.2

Rewrite the expression.

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

Step 16

Multiply $- 1$ by $2$ .

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

Step 17

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

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

Step 18

Simplify the expression.

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Step 18.1

Combine the numerators over the common denominator.

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

Step 18.2

Add $1$ and $1$ .

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

Step 19

Cancel the common factor of $2$ .

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Step 19.1

Cancel the common factor.

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

Step 19.2

Rewrite the expression.

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

Step 20

Simplify.

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

Step 21

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

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

Step 22

Simplify the expression.

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Step 22.1

Multiply $- 1$ by $- 2$ .

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

Step 22.2

Reorder terms.

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