Calculus Examples

$x \ln (x)$

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

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

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

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

Differentiate using the Power Rule.

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Write $x$ as a fraction with denominator $1$ .

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

Simplify terms.

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

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

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

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

$1 + \ln (x)$

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