Calculus Examples

$f (x) = \arctan (\frac{x + a}{b})$

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 $g (x) = \frac{x + a}{b}$ .

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Step 1.1

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

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{x + a}{b}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{x + a}{b}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

By the Sum Rule, the derivative of $x + a$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [a]$ .

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

Step 2.4

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

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

Step 2.5

Since $a$ is constant with respect to $x$ , the derivative of $a$ with respect to $x$ is $0$ .

$\frac{1}{b (1 + {(\frac{x + a}{b})}^{2})} (1 + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

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

Step 2.6.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x + a}{b}$ .

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $b$ by $1$ .

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

Step 3.3.2

Combine $b$ and $\frac{{(x + a)}^{2}}{b^{2}}$ .

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

Step 3.3.3

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

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Step 3.3.3.1

Factor $b$ out of $b {(x + a)}^{2}$ .

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

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

Cancel the common factor.

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

Step 3.3.3.2.3

Rewrite the expression.

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

Step 3.3.4

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

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

Step 3.3.5

Combine the numerators over the common denominator.

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

Step 3.3.6

Raise $b$ to the power of $1$ .

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

Step 3.3.7

Raise $b$ to the power of $1$ .

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

Step 3.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.9

Add $1$ and $1$ .

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

Step 3.3.10

Multiply by the reciprocal of the fraction to divide by $\frac{b^{2} + {(x + a)}^{2}}{b}$ .

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

Step 3.3.11

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

$\frac{b}{b^{2} + {(x + a)}^{2}}$