Calculus Examples

$f (x) = \frac{x^{2} - 4 x}{\sqrt{x}}$

Step 1

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

$\frac{d}{d x} [\frac{x^{2} - 4 x}{x^{\frac{1}{2}}}]$

Step 2

Factor $x$ out of $x^{2} - 4 x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

$\frac{d}{d x} [\frac{x \cdot x - 4 x}{x^{\frac{1}{2}}}]$

Step 2.2

Factor $x$ out of $- 4 x$ .

$\frac{d}{d x} [\frac{x \cdot x + x \cdot - 4}{x^{\frac{1}{2}}}]$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{d}{d x} [\frac{x (x - 4)}{x^{\frac{1}{2}}}]$

Step 3

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

$\frac{d}{d x} [x (x - 4) x^{- \frac{1}{2}}]$

Step 4

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

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Step 4.1

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

$\frac{d}{d x} [x^{- \frac{1}{2}} x (x - 4)]$

Step 4.2

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

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Step 4.2.1

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

$\frac{d}{d x} [x^{- \frac{1}{2}} x^{1} (x - 4)]$

Step 4.2.2

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

$\frac{d}{d x} [x^{- \frac{1}{2} + 1} (x - 4)]$

Step 4.3

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

$\frac{d}{d x} [x^{- \frac{1}{2} + \frac{2}{2}} (x - 4)]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{- 1 + 2}{2}} (x - 4)]$

Step 4.5

Add $- 1$ and $2$ .

$\frac{d}{d x} [x^{\frac{1}{2}} (x - 4)]$

Step 5

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 $g (x) = x - 4$ .

$x^{\frac{1}{2}} \frac{d}{d x} [x - 4] + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$x^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6.2

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

$x^{\frac{1}{2}} (1 + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$x^{\frac{1}{2}} (1 + 0) + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6.4

Simplify the expression.

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Step 6.4.1

Add $1$ and $0$ .

$x^{\frac{1}{2}} \cdot 1 + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6.4.2

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

$x^{\frac{1}{2}} + (x - 4) \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 6.5

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

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 7

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

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 8

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

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 9

Combine the numerators over the common denominator.

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 10.2

Subtract $2$ from $1$ .

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 11

Move the negative in front of the fraction.

$x^{\frac{1}{2}} + (x - 4) (\frac{1}{2} x^{- \frac{1}{2}})$

Step 12

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

$x^{\frac{1}{2}} + (x - 4) \frac{x^{- \frac{1}{2}}}{2}$

Step 13

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

$x^{\frac{1}{2}} + (x - 4) \frac{1}{2 x^{\frac{1}{2}}}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$x^{\frac{1}{2}} + x \frac{1}{2 x^{\frac{1}{2}}} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2

Combine terms.

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Step 14.2.1

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

$x^{\frac{1}{2}} + \frac{x}{2 x^{\frac{1}{2}}} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.2

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

$x^{\frac{1}{2}} + \frac{x \cdot x^{- \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.3

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

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Step 14.2.3.1

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

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Step 14.2.3.1.1

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

$x^{\frac{1}{2}} + \frac{x^{1} x^{- \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.3.1.2

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

$x^{\frac{1}{2}} + \frac{x^{1 - \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.3.2

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

$x^{\frac{1}{2}} + \frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.3.3

Combine the numerators over the common denominator.

$x^{\frac{1}{2}} + \frac{x^{\frac{2 - 1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.3.4

Subtract $1$ from $2$ .

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}}$

Step 14.2.4

Combine $- 4$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 4}{2 x^{\frac{1}{2}}}$

Step 14.2.5

Factor $2$ out of $- 4$ .

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}}$

Step 14.2.6

Cancel the common factors.

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Step 14.2.6.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 14.2.6.2

Cancel the common factor.

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}}$

Step 14.2.6.3

Rewrite the expression.

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} + \frac{- 2}{x^{\frac{1}{2}}}$

Step 14.2.7

Move the negative in front of the fraction.

$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$

Step 14.2.8

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

$x^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$

Step 14.2.9

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

$\frac{x^{\frac{1}{2}} \cdot 2}{2} + \frac{x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$

Step 14.2.10

Combine the numerators over the common denominator.

$\frac{x^{\frac{1}{2}} \cdot 2 + x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$

Step 14.2.11

Move $2$ to the left of $x^{\frac{1}{2}}$ .

$\frac{2 \cdot x^{\frac{1}{2}} + x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$

Step 14.2.12

Add $2 x^{\frac{1}{2}}$ and $x^{\frac{1}{2}}$ .

$\frac{3 x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}}$