Calculus Examples

$f (x) = \frac{\log_{9} (x)}{x^{3}}$

Step 1

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 $f (x) = \log_{9} (x)$ and $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [\log_{9} (x)] - \log_{9} (x) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.1

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

$\frac{x^{3} \frac{d}{d x} [\log_{9} (x)] - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 2.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [\log_{9} (x)] - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3

The derivative of $\log_{9} (x)$ with respect to $x$ is $\frac{1}{x \ln (9)}$ .

$\frac{x^{3} \frac{1}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4

Simplify terms.

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Step 4.1

Combine $x^{3}$ and $\frac{1}{x \ln (9)}$ .

$\frac{\frac{x^{3}}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{3}$ .

$\frac{\frac{x \cdot x^{2}}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x$ out of $x \ln (9)$ .

$\frac{\frac{x \cdot x^{2}}{x (\ln (9))} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.2.2

Cancel the common factor.

$\frac{\frac{x \cdot x^{2}}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.2.3

Rewrite the expression.

$\frac{\frac{x^{2}}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5

Multiply $\frac{\frac{x^{2}}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$ by $\frac{\ln (9)}{\ln (9)}$ .

$\frac{\ln (9)}{\ln (9)} \cdot \frac{\frac{x^{2}}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{\ln (9) (\frac{x^{2}}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{3}])}{\ln (9) x^{6}}$

Step 6.2

Apply the distributive property.

$\frac{\ln (9) \frac{x^{2}}{\ln (9)} + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{3}])}{\ln (9) x^{6}}$

Step 6.3

Cancel the common factor of $\ln (9)$ .

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Step 6.3.1

Cancel the common factor.

$\frac{\ln (9) \frac{x^{2}}{\ln (9)} + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{3}])}{\ln (9) x^{6}}$

Step 6.3.2

Rewrite the expression.

$\frac{x^{2} + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{3}])}{\ln (9) x^{6}}$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{x^{2} + \ln (9) (- \log_{9} (x) (3 x^{2}))}{\ln (9) x^{6}}$

Step 8

Multiply $3$ by $- 1$ .

$\frac{x^{2} + \ln (9) (- 3 \log_{9} (x) x^{2})}{\ln (9) x^{6}}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Simplify $- 3 \log_{9} (x)$ by moving $3$ inside the logarithm.

$\frac{x^{2} + \ln (9) (- \log_{9} (x^{3}) x^{2})}{\ln (9) x^{6}}$

Step 9.1.2

Reorder factors in $x^{2} + \ln (9) (- \log_{9} (x^{3}) x^{2})$ .

$\frac{x^{2} - x^{2} \ln (9) \log_{9} (x^{3})}{\ln (9) x^{6}}$

Step 9.2

Reorder terms.

$\frac{x^{2} - x^{2} \ln (9) \log_{9} (x^{3})}{x^{6} \ln (9)}$

Step 9.3

Factor $x^{2}$ out of $x^{2} - x^{2} \ln (9) \log_{9} (x^{3})$ .

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Step 9.3.1

Multiply by $1$ .

$\frac{x^{2} \cdot 1 - x^{2} \ln (9) \log_{9} (x^{3})}{x^{6} \ln (9)}$

Step 9.3.2

Factor $x^{2}$ out of $- x^{2} \ln (9) \log_{9} (x^{3})$ .

$\frac{x^{2} \cdot 1 + x^{2} (- \ln (9) \log_{9} (x^{3}))}{x^{6} \ln (9)}$

Step 9.3.3

Factor $x^{2}$ out of $x^{2} \cdot 1 + x^{2} (- \ln (9) \log_{9} (x^{3}))$ .

$\frac{x^{2} (1 - \ln (9) \log_{9} (x^{3}))}{x^{6} \ln (9)}$

Step 9.4

Expand $\log_{9} (x^{3})$ by moving $3$ outside the logarithm.

$\frac{x^{2} (1 - \ln (9) (3 \log_{9} (x)))}{x^{6} \ln (9)}$

Step 9.5

Cancel the common factors.

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Step 9.5.1

Factor $x^{2}$ out of $x^{6} \ln (9)$ .

$\frac{x^{2} (1 - \ln (9) (3 \log_{9} (x)))}{x^{2} (x^{4} \ln (9))}$

Step 9.5.2

Cancel the common factor.

$\frac{x^{2} (1 - \ln (9) (3 \log_{9} (x)))}{x^{2} (x^{4} \ln (9))}$

Step 9.5.3

Rewrite the expression.

$\frac{1 - \ln (9) (3 \log_{9} (x))}{x^{4} \ln (9)}$

Step 9.6

Multiply $3$ by $- 1$ .

$\frac{1 - 3 \ln (9) \log_{9} (x)}{x^{4} \ln (9)}$