Calculus Examples

$\ln (x^{2} - 4 x)$

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 $g (x) = x^{2} - 4 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} - 4 x$ .

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2} - 4 x]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} - 4 x$ .

$\frac{1}{x^{2} - 4 x} \frac{d}{d x} [x^{2} - 4 x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 4 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $- 4$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Reorder the factors of $\frac{1}{x^{2} - 4 x} (2 x - 4)$ .

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

Step 3.2

Factor $x$ out of $x^{2} - 4 x$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x$ out of $- 4 x$ .

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

Step 3.2.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$(2 x - 4) \frac{1}{x (x - 4)}$

Step 3.3

Multiply $2 x - 4$ by $\frac{1}{x (x - 4)}$ .

$\frac{2 x - 4}{x (x - 4)}$

Step 3.4

Factor $2$ out of $2 x - 4$ .

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Step 3.4.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{x (x - 4)}$

Step 3.4.2

Factor $2$ out of $- 4$ .

$\frac{2 x + 2 \cdot - 2}{x (x - 4)}$

Step 3.4.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{2 (x - 2)}{x (x - 4)}$