Calculus Examples

$f (x) = x^{2} \ln (x^{2})$

Step 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 $f (x) = x^{2}$ and $g (x) = \ln (x^{2})$ .

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

Step 2

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 $g (x) = x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $\frac{1}{x^{2}}$ and $x^{2}$ .

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

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

Cancel the common factor of $x^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

Multiply $\frac{d}{d x} [x^{2}]$ by $1$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \ln (x^{2}) (2 x)$

Step 3.5

Reorder terms.

$2 x \ln (x^{2}) + 2 x$