Calculus Examples

$2 x \ln (7 x) + x$

Step 1

By the Sum Rule, the derivative of $2 x \ln (7 x) + x$ with respect to $x$ is $\frac{d}{d x} [2 x \ln (7 x)] + \frac{d}{d x} [x]$ .

$\frac{d}{d x} [2 x \ln (7 x)] + \frac{d}{d x} [x]$

Step 2

Evaluate $\frac{d}{d x} [2 x \ln (7 x)]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x \ln (7 x)$ with respect to $x$ is $2 \frac{d}{d x} [x \ln (7 x)]$ .

$2 \frac{d}{d x} [x \ln (7 x)] + \frac{d}{d x} [x]$

Step 2.2

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 $g (x) = \ln (7 x)$ .

$2 (x \frac{d}{d x} [\ln (7 x)] + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.3

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 $g (x) = 7 x$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $7 x$ .

$2 (x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [7 x]) + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.3.2

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

$2 (x (\frac{1}{u} \frac{d}{d x} [7 x]) + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.3.3

Replace all occurrences of $u$ with $7 x$ .

$2 (x (\frac{1}{7 x} \frac{d}{d x} [7 x]) + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.4

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$2 (x (\frac{1}{7 x} (7 \frac{d}{d x} [x])) + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.5

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

$2 (x (\frac{1}{7 x} (7 \cdot 1)) + \ln (7 x) \frac{d}{d x} [x]) + \frac{d}{d x} [x]$

Step 2.6

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

$2 (x (\frac{1}{7 x} (7 \cdot 1)) + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.7

Multiply $7$ by $1$ .

$2 (x (\frac{1}{7 x} \cdot 7) + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.8

Combine $\frac{1}{7 x}$ and $7$ .

$2 (x \frac{7}{7 x} + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.9

Cancel the common factor of $7$ .

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Step 2.9.1

Cancel the common factor.

$2 (x \frac{7}{7 x} + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.9.2

Rewrite the expression.

$2 (x \frac{1}{x} + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.10

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

$2 (\frac{x}{x} + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.11

Cancel the common factor of $x$ .

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Step 2.11.1

Cancel the common factor.

$2 (\frac{x}{x} + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.11.2

Rewrite the expression.

$2 (1 + \ln (7 x) \cdot 1) + \frac{d}{d x} [x]$

Step 2.12

Multiply $\ln (7 x)$ by $1$ .

$2 (1 + \ln (7 x)) + \frac{d}{d x} [x]$

Step 3

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

$2 (1 + \ln (7 x)) + 1$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 \cdot 1 + 2 \ln (7 x) + 1$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $2$ by $1$ .

$2 + 2 \ln (7 x) + 1$

Step 4.2.2

Add $2$ and $1$ .

$2 \ln (7 x) + 3$