Calculus Examples

$\arctan (\frac{x}{y})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{y}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{x}{y}]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{y}$ .

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

Step 2

Differentiate.

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Step 2.1

Since $\frac{1}{y}$ is constant with respect to $x$ , the derivative of $\frac{x}{y}$ with respect to $x$ is $\frac{1}{y} \frac{d}{d x} [x]$ .

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

Step 2.2

Multiply $\frac{1}{y}$ by $\frac{1}{1 + {(\frac{x}{y})}^{2}}$ .

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

Step 2.3

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

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

Step 2.4

Multiply $\frac{1}{y (1 + {(\frac{x}{y})}^{2})}$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{y}$ .

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

Step 3.2

Apply the distributive property.

$\frac{1}{y \cdot 1 + y \frac{x^{2}}{y^{2}}}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $y$ by $1$ .

$\frac{1}{y + y \frac{x^{2}}{y^{2}}}$

Step 3.3.2

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

$\frac{1}{y + \frac{y x^{2}}{y^{2}}}$

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

Cancel the common factor.

$\frac{1}{y + \frac{y x^{2}}{y \cdot y}}$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{1}{y + \frac{x^{2}}{y}}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

To write $y$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{1}{\frac{y \cdot y}{y} + \frac{x^{2}}{y}}$

Step 3.4.2

Combine the numerators over the common denominator.

$\frac{1}{\frac{y \cdot y + x^{2}}{y}}$

Step 3.4.3

Multiply $y$ by $y$ .

$\frac{1}{\frac{y^{2} + x^{2}}{y}}$

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{y}{y^{2} + x^{2}}$

Step 3.6

Multiply $\frac{y}{y^{2} + x^{2}}$ by $1$ .

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