Calculus Examples

$2 x \ln (2 x)$

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (2 x)$ .

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

Step 3

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 $g (x) = 2 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 3.2

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

$2 (x (\frac{1}{u} \frac{d}{d x} [2 x]) + \ln (2 x) \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u$ with $2 x$ .

$2 (x (\frac{1}{2 x} \frac{d}{d x} [2 x]) + \ln (2 x) \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$2 (\frac{1}{2} \frac{d}{d x} [2 x] + \ln (2 x) \frac{d}{d x} [x])$

Step 4.3

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

$2 (\frac{1}{2} (2 \frac{d}{d x} [x]) + \ln (2 x) \frac{d}{d x} [x])$

Step 4.4

Simplify terms.

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor of $2$ .

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Rewrite the expression.

$2 (1 \frac{d}{d x} [x] + \ln (2 x) \frac{d}{d x} [x])$

Step 4.4.3

Multiply $\frac{d}{d x} [x]$ by $1$ .

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

Step 4.5

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

$2 (1 + \ln (2 x) \frac{d}{d x} [x])$

Step 4.6

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

$2 (1 + \ln (2 x) \cdot 1)$

Step 4.7

Multiply $\ln (2 x)$ by $1$ .

$2 (1 + \ln (2 x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$2 \cdot 1 + 2 \ln (2 x)$

Step 5.2

Multiply $2$ by $1$ .

$2 + 2 \ln (2 x)$

Step 5.3

Reorder terms.

$2 \ln (2 x) + 2$