Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2} y]$

Step 1.3

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

$\frac{1}{x^{2} y} \frac{d}{d x} [x^{2} y]$

Step 2

Differentiate.

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Step 2.1

Since $y$ is constant with respect to $x$ , the derivative of $x^{2} y$ with respect to $x$ is $y \frac{d}{d x} [x^{2}]$ .

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

Step 2.2

Simplify terms.

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Step 2.2.1

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

$\frac{y}{x^{2} y} \frac{d}{d x} [x^{2}]$

Step 2.2.2

Cancel the common factor of $y$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{y}{x^{2} y} \frac{d}{d x} [x^{2}]$

Step 2.2.2.2

Rewrite the expression.

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

Step 2.3

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

$\frac{1}{x^{2}} (2 x)$

Step 2.4

Simplify terms.

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Step 2.4.1

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

$\frac{2}{x^{2}} x$

Step 2.4.2

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

$\frac{2 x}{x^{2}}$

Step 2.4.3

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

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Step 2.4.3.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{x^{2}}$

Step 2.4.3.2

Cancel the common factors.

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Step 2.4.3.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot 2}{x \cdot x}$

Step 2.4.3.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 2.4.3.2.3

Rewrite the expression.

$\frac{2}{x}$