Calculus Examples

$\frac{\ln (3 x)}{2 x^{2}}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{\ln (3 x)}{2 x^{2}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{\ln (3 x)}{x^{2}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{\ln (3 x)}{x^{2}}]$

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 $f (x) = \ln (3 x)$ and $g (x) = x^{2}$ .

$\frac{1}{2} \cdot \frac{x^{2} \frac{d}{d x} [\ln (3 x)] - \ln (3 x) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3

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

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Step 3.1

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

$\frac{1}{2} \cdot \frac{x^{2} \frac{d}{d x} [\ln (3 x)] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{x^{2} \frac{d}{d x} [\ln (3 x)] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{1}{2} \cdot \frac{x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]) - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

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

$\frac{1}{2} \cdot \frac{x^{2} (\frac{1}{u} \frac{d}{d x} [3 x]) - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{2} \cdot \frac{x^{2} (\frac{1}{3 x} \frac{d}{d x} [3 x]) - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{3 x}$ and $x^{2}$ .

$\frac{1}{2} \cdot \frac{\frac{x^{2}}{3 x} \frac{d}{d x} [3 x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{1}{2} \cdot \frac{\frac{x \cdot x}{3 x} \frac{d}{d x} [3 x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $x$ out of $3 x$ .

$\frac{1}{2} \cdot \frac{\frac{x \cdot x}{x \cdot 3} \frac{d}{d x} [3 x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{\frac{x \cdot x}{x \cdot 3} \frac{d}{d x} [3 x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{\frac{x}{3} \frac{d}{d x} [3 x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{2} \cdot \frac{\frac{x}{3} (3 \frac{d}{d x} [x]) - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.4

Simplify terms.

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Step 5.4.1

Combine $3$ and $\frac{x}{3}$ .

$\frac{1}{2} \cdot \frac{\frac{3 x}{3} \frac{d}{d x} [x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.4.2

Cancel the common factor of $3$ .

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Step 5.4.2.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{\frac{3 x}{3} \frac{d}{d x} [x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.4.2.2

Divide $x$ by $1$ .

$\frac{1}{2} \cdot \frac{x \frac{d}{d x} [x] - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.5

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

$\frac{1}{2} \cdot \frac{x \cdot 1 - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.6

Multiply $x$ by $1$ .

$\frac{1}{2} \cdot \frac{x - \ln (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.7

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

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

Step 5.8

Simplify with factoring out.

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Step 5.8.1

Multiply $2$ by $- 1$ .

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

Step 5.8.2

Factor $x$ out of $x - 2 \ln (3 x) x$ .

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Step 5.8.2.1

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

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

Step 5.8.2.2

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

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

Step 5.8.2.3

Factor $x$ out of $- 2 \ln (3 x) x$ .

$\frac{1}{2} \cdot \frac{x \cdot 1 + x (- 2 \ln (3 x))}{x^{4}}$

Step 5.8.2.4

Factor $x$ out of $x \cdot 1 + x (- 2 \ln (3 x))$ .

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

Step 6

Cancel the common factors.

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Step 6.1

Factor $x$ out of $x^{4}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\frac{1}{2}$ by $\frac{1 - 2 \ln (3 x)}{x^{3}}$ .

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

Step 8

Simplify each term.

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Step 8.1

Simplify $- 2 \ln (3 x)$ by moving $2$ inside the logarithm.

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

Step 8.2

Apply the product rule to $3 x$ .

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

Step 8.3

Raise $3$ to the power of $2$ .

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