Calculus Examples

$x \ln (x)$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $x$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.3.3

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

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

Step 1.3.4

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

$f' (x) = 1 + \ln (x)$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $1 + \ln (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\ln (x)]$ .

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

Step 2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\ln (x)]$

Step 2.2

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

$0 + \frac{1}{x}$

Step 2.3

Add $0$ and $\frac{1}{x}$ .

$f'' (x) = \frac{1}{x}$