Calculus Examples

$y = \arctan (\frac{x}{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan (\frac{x}{2}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = \frac{x}{2}$ .

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Step 3.1.1

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

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{x}{2}]$

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

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Step 3.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the product rule to $\frac{x}{2}$ .

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Combine terms.

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Step 3.3.3.1

Multiply $2$ by $1$ .

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

Step 3.3.3.2

Raise $2$ to the power of $2$ .

$\frac{1}{2 + 2 \frac{x^{2}}{4}}$

Step 3.3.3.3

Combine $2$ and $\frac{x^{2}}{4}$ .

$\frac{1}{2 + \frac{2 x^{2}}{4}}$

Step 3.3.3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.3.3.4.1

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

$\frac{1}{2 + \frac{2 (x^{2})}{4}}$

Step 3.3.3.4.2

Cancel the common factors.

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Step 3.3.3.4.2.1

Factor $2$ out of $4$ .

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

Step 3.3.3.4.2.2

Cancel the common factor.

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

Step 3.3.3.4.2.3

Rewrite the expression.

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

Step 3.3.4

Reorder terms.

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

Step 3.3.5

Simplify the denominator.

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Step 3.3.5.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3.5.2

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

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

Step 3.3.5.3

Combine the numerators over the common denominator.

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

Step 3.3.5.4

Multiply $2$ by $2$ .

$\frac{1}{\frac{x^{2} + 4}{2}}$

Step 3.3.6

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2}{x^{2} + 4}$

Step 3.3.7

Multiply $\frac{2}{x^{2} + 4}$ by $1$ .

$\frac{2}{x^{2} + 4}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2}{x^{2} + 4}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2}{x^{2} + 4}$