Calculus Examples

$\arctan (x - 1)$

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 $g (x) = x - 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [x - 1]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

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

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

Step 3

Simplify.

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Step 3.1

Reorder terms.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$\frac{1}{(x - 1) (x - 1) + 1}$

Step 3.2.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$\frac{1}{x (x - 1) - 1 (x - 1) + 1}$

Step 3.2.2.2

Apply the distributive property.

$\frac{1}{x \cdot x + x \cdot - 1 - 1 (x - 1) + 1}$

Step 3.2.2.3

Apply the distributive property.

$\frac{1}{x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 1}$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $x$ by $x$ .

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

Step 3.2.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 3.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 3.2.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 3.2.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2.3.2

Subtract $x$ from $- x$ .

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

Step 3.2.4

Add $1$ and $1$ .

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