Calculus Examples

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

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Simplify terms.

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Step 4.2.1

Combine $\frac{2}{x}$ and $\ln (x)$ .

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

Step 4.2.2

Combine $x$ and $\frac{2 \ln (x)}{x}$ .

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

Step 4.2.3

Cancel the common factor of $x$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Divide $2 \ln (x)$ by $1$ .

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

Step 4.3

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

$2 \ln (x) + \ln^{2} (x) \cdot 1$

Step 4.4

Simplify the expression.

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Step 4.4.1

Multiply $\ln^{2} (x)$ by $1$ .

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

Step 4.4.2

Reorder terms.

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