Calculus Examples

$f (x) = \frac{\frac{1}{3}}{4 - 9 \frac{1}{3}}$

Step 1

Find the first derivative.

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Step 1.1

Convert $9 \frac{1}{3}$ to an improper fraction.

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Step 1.1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - (9 + \frac{1}{3})}]$

Step 1.1.2

Add $9$ and $\frac{1}{3}$ .

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Step 1.1.2.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - (9 \cdot \frac{3}{3} + \frac{1}{3})}]$

Step 1.1.2.2

Combine $9$ and $\frac{3}{3}$ .

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - (\frac{9 \cdot 3}{3} + \frac{1}{3})}]$

Step 1.1.2.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - \frac{9 \cdot 3 + 1}{3}}]$

Step 1.1.2.4

Simplify the numerator.

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Step 1.1.2.4.1

Multiply $9$ by $3$ .

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - \frac{27 + 1}{3}}]$

Step 1.1.2.4.2

Add $27$ and $1$ .

$\frac{d}{d x} [\frac{\frac{1}{3}}{4 - \frac{28}{3}}]$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{4 - \frac{28}{3}}]$

Step 1.3

Simplify the denominator.

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Step 1.3.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{4 \cdot \frac{3}{3} - \frac{28}{3}}]$

Step 1.3.2

Combine $4$ and $\frac{3}{3}$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{\frac{4 \cdot 3}{3} - \frac{28}{3}}]$

Step 1.3.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{\frac{4 \cdot 3 - 28}{3}}]$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Multiply $4$ by $3$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{\frac{12 - 28}{3}}]$

Step 1.3.4.2

Subtract $28$ from $12$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{\frac{- 16}{3}}]$

Step 1.3.5

Move the negative in front of the fraction.

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{1}{- \frac{16}{3}}]$

Step 1.4

Cancel the common factor of $1$ and $- 1$ .

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Step 1.4.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{- 1 (- 1)}{- \frac{16}{3}}]$

Step 1.4.2

Move the negative in front of the fraction.

$\frac{d}{d x} [\frac{1}{3} (- \frac{1}{\frac{16}{3}})]$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{d}{d x} [\frac{1}{3} (- (1 (\frac{3}{16})))]$

Step 1.6

Reduce the expression by cancelling the common factors.

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Step 1.6.1

Multiply $\frac{3}{16}$ by $1$ .

$\frac{d}{d x} [\frac{1}{3} (- \frac{3}{16})]$

Step 1.6.2

Cancel the common factor of $3$ .

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Step 1.6.2.1

Move the leading negative in $- \frac{3}{16}$ into the numerator.

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{- 3}{16}]$

Step 1.6.2.2

Factor $3$ out of $- 3$ .

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{3 (- 1)}{16}]$

Step 1.6.2.3

Cancel the common factor.

$\frac{d}{d x} [\frac{1}{3} \cdot \frac{3 \cdot - 1}{16}]$

Step 1.6.2.4

Rewrite the expression.

$\frac{d}{d x} [\frac{- 1}{16}]$

Step 1.6.3

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{1}{16}]$

Step 1.7

Since $- \frac{1}{16}$ is constant with respect to $x$ , the derivative of $- \frac{1}{16}$ with respect to $x$ is $0$ .

$f' (x) = 0$

Step 2

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

$f'' (x) = 0$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $0$ .

$0$