Calculus Examples

$f (x) = \frac{10 \log_{4} (x)}{x}$

Step 1

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10 \log_{4} (x)}{x}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{\log_{4} (x)}{x}]$ .

$10 \frac{d}{d x} [\frac{\log_{4} (x)}{x}]$

Step 2

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 $f (x) = \log_{4} (x)$ and $g (x) = x$ .

$10 \frac{x \frac{d}{d x} [\log_{4} (x)] - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

The derivative of $\log_{4} (x)$ with respect to $x$ is $\frac{1}{x \ln (4)}$ .

$10 \frac{x \frac{1}{x \ln (4)} - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine $x$ and $\frac{1}{x \ln (4)}$ .

$10 \frac{\frac{x}{x \ln (4)} - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

$10 \frac{\frac{x}{x \ln (4)} - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2.2

Rewrite the expression.

$10 \frac{\frac{1}{\ln (4)} - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}}$

Step 5

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

$10 (\frac{\ln (4)}{\ln (4)} \cdot \frac{\frac{1}{\ln (4)} - \log_{4} (x) \frac{d}{d x} [x]}{x^{2}})$

Step 6

Simplify terms.

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Step 6.1

Combine.

$10 \frac{\ln (4) (\frac{1}{\ln (4)} - \log_{4} (x) \frac{d}{d x} [x])}{\ln (4) x^{2}}$

Step 6.2

Apply the distributive property.

$10 \frac{\ln (4) \frac{1}{\ln (4)} + \ln (4) (- \log_{4} (x) \frac{d}{d x} [x])}{\ln (4) x^{2}}$

Step 6.3

Cancel the common factor of $\ln (4)$ .

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Step 6.3.1

Cancel the common factor.

$10 \frac{\ln (4) \frac{1}{\ln (4)} + \ln (4) (- \log_{4} (x) \frac{d}{d x} [x])}{\ln (4) x^{2}}$

Step 6.3.2

Rewrite the expression.

$10 \frac{1 + \ln (4) (- \log_{4} (x) \frac{d}{d x} [x])}{\ln (4) x^{2}}$

Step 7

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

$10 \frac{1 + \ln (4) (- \log_{4} (x) \cdot 1)}{\ln (4) x^{2}}$

Step 8

Combine fractions.

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Step 8.1

Multiply $- 1$ by $1$ .

$10 \frac{1 + \ln (4) (- \log_{4} (x))}{\ln (4) x^{2}}$

Step 8.2

Combine $10$ and $\frac{1 + \ln (4) (- \log_{4} (x))}{\ln (4) x^{2}}$ .

$\frac{10 (1 + \ln (4) (- \log_{4} (x)))}{\ln (4) x^{2}}$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$\frac{10 \cdot 1 + 10 (\ln (4) (- \log_{4} (x)))}{\ln (4) x^{2}}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $10$ by $1$ .

$\frac{10 + 10 (\ln (4) (- \log_{4} (x)))}{\ln (4) x^{2}}$

Step 9.2.1.2

Multiply $10 (\ln (4) (- \log_{4} (x)))$ .

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Step 9.2.1.2.1

Reorder $\ln (4)$ and $10$ .

$\frac{10 + 10 \cdot \ln (4) (- \log_{4} (x))}{\ln (4) x^{2}}$

Step 9.2.1.2.2

Simplify $10 \cdot \ln (4)$ by moving $10$ inside the logarithm.

$\frac{10 + \ln (4^{10}) (- \log_{4} (x))}{\ln (4) x^{2}}$

Step 9.2.1.3

Raise $4$ to the power of $10$ .

$\frac{10 + \ln (1048576) (- \log_{4} (x))}{\ln (4) x^{2}}$

Step 9.2.2

Reorder factors in $10 + \ln (1048576) (- \log_{4} (x))$ .

$\frac{10 - \ln (1048576) \log_{4} (x)}{\ln (4) x^{2}}$

Step 9.3

Reorder terms.

$\frac{- \ln (1048576) \log_{4} (x) + 10}{x^{2} \ln (4)}$