Calculus Examples

$f (x) = 5 x \ln (3 x) - 5 x$

Step 1

By the Sum Rule, the derivative of $5 x \ln (3 x) - 5 x$ with respect to $x$ is $\frac{d}{d x} [5 x \ln (3 x)] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [5 x \ln (3 x)] + \frac{d}{d x} [- 5 x]$

Step 2

Evaluate $\frac{d}{d x} [5 x \ln (3 x)]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x \ln (3 x)$ with respect to $x$ is $5 \frac{d}{d x} [x \ln (3 x)]$ .

$5 \frac{d}{d x} [x \ln (3 x)] + \frac{d}{d x} [- 5 x]$

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 $g (x) = \ln (3 x)$ .

$5 (x \frac{d}{d x} [\ln (3 x)] + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

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 $g (x) = 3 x$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$5 (x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

Step 2.3.2

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

$5 (x (\frac{1}{u} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

Step 2.3.3

Replace all occurrences of $u$ with $3 x$ .

$5 (x (\frac{1}{3 x} \frac{d}{d x} [3 x]) + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

Step 2.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$5 (x (\frac{1}{3 x} (3 \frac{d}{d x} [x])) + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

Step 2.5

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

$5 (x (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 5 x]$

Step 2.6

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

$5 (x (\frac{1}{3 x} (3 \cdot 1)) + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.7

Multiply $3$ by $1$ .

$5 (x (\frac{1}{3 x} \cdot 3) + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.8

Combine $\frac{1}{3 x}$ and $3$ .

$5 (x \frac{3}{3 x} + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.9

Cancel the common factor of $3$ .

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Step 2.9.1

Cancel the common factor.

$5 (x \frac{3}{3 x} + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.9.2

Rewrite the expression.

$5 (x \frac{1}{x} + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.10

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

$5 (\frac{x}{x} + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.11

Cancel the common factor of $x$ .

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Step 2.11.1

Cancel the common factor.

$5 (\frac{x}{x} + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.11.2

Rewrite the expression.

$5 (1 + \ln (3 x) \cdot 1) + \frac{d}{d x} [- 5 x]$

Step 2.12

Multiply $\ln (3 x)$ by $1$ .

$5 (1 + \ln (3 x)) + \frac{d}{d x} [- 5 x]$

Step 3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$5 (1 + \ln (3 x)) - 5 \frac{d}{d x} [x]$

Step 3.2

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

$5 (1 + \ln (3 x)) - 5 \cdot 1$

Step 3.3

Multiply $- 5$ by $1$ .

$5 (1 + \ln (3 x)) - 5$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$5 \cdot 1 + 5 \ln (3 x) - 5$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $5$ by $1$ .

$5 + 5 \ln (3 x) - 5$

Step 4.2.2

Subtract $5$ from $5$ .

$5 \ln (3 x) + 0$

Step 4.2.3

Add $5 \ln (3 x)$ and $0$ .

$5 \ln (3 x)$