Calculus Examples

$f (x) = \frac{\log_{9} (x)}{x^{2}}$

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 $g (x) = x^{2}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [\log_{9} (x)] - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Combine $x^{2}$ and $\frac{1}{x \ln (9)}$ .

$\frac{\frac{x^{2}}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.2.1

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

$\frac{\frac{x \cdot x}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\frac{x \cdot x}{x (\ln (9))} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2.2.2

Cancel the common factor.

$\frac{\frac{x \cdot x}{x \ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2.2.3

Rewrite the expression.

$\frac{\frac{x}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

Multiply $\frac{\frac{x}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$ by $\frac{\ln (9)}{\ln (9)}$ .

$\frac{\ln (9)}{\ln (9)} \cdot \frac{\frac{x}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{\ln (9) (\frac{x}{\ln (9)} - \log_{9} (x) \frac{d}{d x} [x^{2}])}{\ln (9) x^{4}}$

Step 6.2

Apply the distributive property.

$\frac{\ln (9) \frac{x}{\ln (9)} + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{2}])}{\ln (9) x^{4}}$

Step 6.3

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

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Step 6.3.1

Cancel the common factor.

$\frac{\ln (9) \frac{x}{\ln (9)} + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{2}])}{\ln (9) x^{4}}$

Step 6.3.2

Rewrite the expression.

$\frac{x + \ln (9) (- \log_{9} (x) \frac{d}{d x} [x^{2}])}{\ln (9) x^{4}}$

Step 7

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

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

Step 8

Multiply $2$ by $- 1$ .

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

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Simplify $- 2 \log_{9} (x)$ by moving $2$ inside the logarithm.

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

Step 9.1.2

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

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

Step 9.2

Reorder terms.

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

Factor $x$ out of $x^{1}$ .

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

Step 9.3.4

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

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

Step 9.4

Expand $\log_{9} (x^{2})$ by moving $2$ outside the logarithm.

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

Step 9.5

Cancel the common factors.

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Step 9.5.1

Factor $x$ out of $x^{4} \ln (9)$ .

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

Step 9.5.2

Cancel the common factor.

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

Step 9.5.3

Rewrite the expression.

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

Step 9.6

Multiply $2$ by $- 1$ .

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

Step 9.7

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

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

Step 9.8

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

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

Step 9.9

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

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

Step 9.10

Rewrite $- (2 \ln (9) \log_{9} (x) - 1)$ as $- 1 (2 \ln (9) \log_{9} (x) - 1)$ .

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

Step 9.11

Move the negative in front of the fraction.

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