Calculus Examples

$f (x) = \frac{6 x}{\ln (x)}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{\ln (x)}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{\ln (x)}]$ .

$6 \frac{d}{d x} [\frac{x}{\ln (x)}]$

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 $g (x) = \ln (x)$ .

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

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 $n = 1$ .

$6 \frac{\ln (x) \cdot 1 - x \frac{d}{d x} [\ln (x)]}{\ln^{2} (x)}$

Step 3.2

Multiply $\ln (x)$ by $1$ .

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Combine $\frac{1}{x}$ and $x$ .

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Combine $6$ and $\frac{\ln (x) - 1}{\ln^{2} (x)}$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify each term.

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Step 6.2.1

Simplify $6 \ln (x)$ by moving $6$ inside the logarithm.

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

Step 6.2.2

Multiply $6$ by $- 1$ .

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