Calculus Examples

$\frac{\sqrt{x} (x - 3)}{3}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} (x - 3)}{3}]$

Step 1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{\frac{1}{2}} (x - 3)}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{\frac{1}{2}} (x - 3)]$ .

$\frac{1}{3} \frac{d}{d x} [x^{\frac{1}{2}} (x - 3)]$

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 $g (x) = x - 3$ .

$\frac{1}{3} (x^{\frac{1}{2}} \frac{d}{d x} [x - 3] + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{1}{3} (x^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.2

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

$\frac{1}{3} (x^{\frac{1}{2}} (1 + \frac{d}{d x} [- 3]) + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{1}{3} (x^{\frac{1}{2}} (1 + 0) + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$\frac{1}{3} (x^{\frac{1}{2}} \cdot 1 + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.4.2

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

$\frac{1}{3} (x^{\frac{1}{2}} + (x - 3) \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 3.5

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Combine terms.

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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Step 11.3.3.1

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

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Step 11.3.3.1.1

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

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

Step 11.3.3.1.2

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

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

Step 11.3.3.2

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

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

Step 11.3.3.3

Combine the numerators over the common denominator.

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

Step 11.3.3.4

Subtract $1$ from $2$ .

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

Step 11.3.4

Multiply $\frac{1}{3}$ by $\frac{x^{\frac{1}{2}}}{2}$ .

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

Step 11.3.5

Multiply $3$ by $2$ .

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

Step 11.3.6

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

$\frac{1}{3} x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{6} + \frac{1}{3} \cdot \frac{- 3}{2 x^{\frac{1}{2}}}$

Step 11.3.7

Move the negative in front of the fraction.

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

Step 11.3.8

Multiply $\frac{1}{3}$ by $\frac{3}{2 x^{\frac{1}{2}}}$ .

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

Step 11.3.9

Multiply $2$ by $3$ .

$\frac{1}{3} x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{6} - \frac{3}{6 x^{\frac{1}{2}}}$

Step 11.3.10

Factor $3$ out of $3$ .

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

Step 11.3.11

Cancel the common factors.

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Step 11.3.11.1

Factor $3$ out of $6 x^{\frac{1}{2}}$ .

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

Step 11.3.11.2

Cancel the common factor.

$\frac{1}{3} x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{6} - \frac{3 \cdot 1}{3 (2 x^{\frac{1}{2}})}$

Step 11.3.11.3

Rewrite the expression.

$\frac{1}{3} x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.12

To write $\frac{1}{3} x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{1}{3} x^{\frac{1}{2}} \cdot \frac{6}{6} + \frac{x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.13

Combine $\frac{1}{3} x^{\frac{1}{2}}$ and $\frac{6}{6}$ .

$\frac{\frac{1}{3} x^{\frac{1}{2}} \cdot 6}{6} + \frac{x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.14

Combine the numerators over the common denominator.

$\frac{\frac{1}{3} x^{\frac{1}{2}} \cdot 6 + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.15

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

$\frac{\frac{x^{\frac{1}{2}}}{3} \cdot 6 + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.16

Combine $\frac{x^{\frac{1}{2}}}{3}$ and $6$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot 6}{3} + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.17

Move $6$ to the left of $x^{\frac{1}{2}}$ .

$\frac{\frac{6 \cdot x^{\frac{1}{2}}}{3} + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.18

Factor $3$ out of $6 x^{\frac{1}{2}}$ .

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

Step 11.3.19

Cancel the common factors.

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Step 11.3.19.1

Factor $3$ out of $3$ .

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

Step 11.3.19.2

Cancel the common factor.

$\frac{\frac{3 (2 x^{\frac{1}{2}})}{3 \cdot 1} + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.19.3

Rewrite the expression.

$\frac{\frac{2 x^{\frac{1}{2}}}{1} + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.19.4

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

$\frac{2 x^{\frac{1}{2}} + x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.20

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

$\frac{3 x^{\frac{1}{2}}}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.21

Factor $3$ out of $3 x^{\frac{1}{2}}$ .

$\frac{3 (x^{\frac{1}{2}})}{6} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 11.3.22

Cancel the common factors.

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Step 11.3.22.1

Factor $3$ out of $6$ .

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

Step 11.3.22.2

Cancel the common factor.

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

Step 11.3.22.3

Rewrite the expression.

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