Calculus Examples

$y = \ln (9 x)$

Step 1

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 $g (x) = 9 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9 x$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $9 x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Simplify terms.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor of $9$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

$\frac{1}{x}$