Calculus Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Factor $3$ out of $3 x$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $3 (x)$ .

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

Step 1.2.2

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

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

Step 2

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 $g (x) = 9 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $9 x^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [9 x^{2}]$

Step 2.2

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

$\frac{1}{u} \frac{d}{d x} [9 x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $9 x^{2}$ .

$\frac{1}{9 x^{2}} \frac{d}{d x} [9 x^{2}]$

Step 3

Differentiate.

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Step 3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{9 x^{2}} (9 \frac{d}{d x} [x^{2}])$

Step 3.2

Simplify terms.

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Step 3.2.1

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

$\frac{9}{9 x^{2}} \frac{d}{d x} [x^{2}]$

Step 3.2.2

Cancel the common factor of $9$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{9}{9 x^{2}} \frac{d}{d x} [x^{2}]$

Step 3.2.2.2

Rewrite the expression.

$\frac{1}{x^{2}} \frac{d}{d x} [x^{2}]$

Step 3.3

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

$\frac{1}{x^{2}} (2 x)$

Step 3.4

Simplify terms.

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Step 3.4.1

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

$\frac{2}{x^{2}} x$

Step 3.4.2

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

$\frac{2 x}{x^{2}}$

Step 3.4.3

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

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Step 3.4.3.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{x^{2}}$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

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

$\frac{x \cdot 2}{x \cdot x}$

Step 3.4.3.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 3.4.3.2.3

Rewrite the expression.

$\frac{2}{x}$