Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $x$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

$1 + \ln (x) - 1$

Step 4

Combine terms.

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Step 4.1

Subtract $1$ from $1$ .

$0 + \ln (x)$

Step 4.2

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

$\ln (x)$