Calculus Examples

$f (x) = x - \ln (8 x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x - \ln (8 x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \ln (8 x)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \ln (8 x)]$

Step 1.2

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

$1 + \frac{d}{d x} [- \ln (8 x)]$

Step 2

Evaluate $\frac{d}{d x} [- \ln (8 x)]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (8 x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (8 x)]$ .

$1 - \frac{d}{d x} [\ln (8 x)]$

Step 2.2

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 $g (x) = 8 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $8 x$ .

$1 - (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [8 x])$

Step 2.2.2

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

$1 - (\frac{1}{u} \frac{d}{d x} [8 x])$

Step 2.2.3

Replace all occurrences of $u$ with $8 x$ .

$1 - (\frac{1}{8 x} \frac{d}{d x} [8 x])$

Step 2.3

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$1 - (\frac{1}{8 x} (8 \frac{d}{d x} [x]))$

Step 2.4

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

$1 - (\frac{1}{8 x} (8 \cdot 1))$

Step 2.5

Multiply $8$ by $1$ .

$1 - (\frac{1}{8 x} \cdot 8)$

Step 2.6

Combine $\frac{1}{8 x}$ and $8$ .

$1 - \frac{8}{8 x}$

Step 2.7

Cancel the common factor of $8$ .

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Step 2.7.1

Cancel the common factor.

$1 - \frac{8}{8 x}$

Step 2.7.2

Rewrite the expression.

$1 - \frac{1}{x}$

Step 3

Reorder terms.

$- \frac{1}{x} + 1$