Calculus Examples

$y = \frac{4 x}{\log_{4} (x)}$

Step 1

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

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

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

Multiply $\log_{4} (x)$ by $1$ .

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify each term.

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Step 6.2.1

Simplify $4 \log_{4} (x)$ by moving $4$ inside the logarithm.

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

Step 6.2.2

Multiply $4 (- \frac{1}{\ln (4)})$ .

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Step 6.2.2.1

Multiply $- 1$ by $4$ .

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

Step 6.2.2.2

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

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

Step 6.2.3

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 6.2.4

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

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

Step 6.2.5

Cancel the common factor of $- 4$ and $2$ .

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Step 6.2.5.1

Factor $2$ out of $- 4$ .

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

Step 6.2.5.2

Cancel the common factors.

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Step 6.2.5.2.1

Factor $2$ out of $2 \ln (2)$ .

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

Step 6.2.5.2.2

Cancel the common factor.

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

Step 6.2.5.2.3

Rewrite the expression.

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

Step 6.2.6

Move the negative in front of the fraction.

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