Calculus Examples

$y = 9 \ln (6 x)$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 \ln (6 x)$ with respect to $x$ is $9 \frac{d}{d x} [\ln (6 x)]$ .

$9 \frac{d}{d x} [\ln (6 x)]$

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 $g (x) = 6 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x$ .

$9 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [6 x])$

Step 2.2

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

$9 (\frac{1}{u} \frac{d}{d x} [6 x])$

Step 2.3

Replace all occurrences of $u$ with $6 x$ .

$9 (\frac{1}{6 x} \frac{d}{d x} [6 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{6 x}$ and $9$ .

$\frac{9}{6 x} \frac{d}{d x} [6 x]$

Step 3.2

Cancel the common factor of $9$ and $6$ .

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Step 3.2.1

Factor $3$ out of $9$ .

$\frac{3 (3)}{6 x} \frac{d}{d x} [6 x]$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $3$ out of $6 x$ .

$\frac{3 (3)}{3 (2 x)} \frac{d}{d x} [6 x]$

Step 3.2.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 (2 x)} \frac{d}{d x} [6 x]$

Step 3.2.2.3

Rewrite the expression.

$\frac{3}{2 x} \frac{d}{d x} [6 x]$

Step 3.3

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

$\frac{3}{2 x} (6 \frac{d}{d x} [x])$

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $6$ and $\frac{3}{2 x}$ .

$\frac{6 \cdot 3}{2 x} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $6$ by $3$ .

$\frac{18}{2 x} \frac{d}{d x} [x]$

Step 3.4.3

Cancel the common factor of $18$ and $2$ .

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Step 3.4.3.1

Factor $2$ out of $18$ .

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

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Factor $2$ out of $2 x$ .

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

Step 3.4.3.2.2

Cancel the common factor.

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

Step 3.4.3.2.3

Rewrite the expression.

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

Step 3.5

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

$\frac{9}{x} \cdot 1$

Step 3.6

Multiply $\frac{9}{x}$ by $1$ .

$\frac{9}{x}$