Calculus Examples

$\arctan (x^{2} - 2 x)$

Step 1

Write $\arctan (x^{2} - 2 x)$ as a function.

$f (x) = \arctan (x^{2} - 2 x)$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $x^{2} - 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x]$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

Simplify the expression.

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Step 2.1.2.5.1

Multiply $- 2$ by $1$ .

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

Step 2.1.2.5.2

Reorder the factors of $\frac{1}{1 + {(x^{2} - 2 x)}^{2}} (2 x - 2)$ .

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

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $(2 x - 2) (\frac{1}{1 + {(x^{2} - 2 x)}^{2}})$ .

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

Step 3

Set the first derivative equal to $0$ then solve the equation $(2 x - 2) (\frac{1}{1 + {(x^{2} - 2 x)}^{2}}) = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 2 = 0$

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

Step 3.3

Set $2 x - 2$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $2 x - 2$ equal to $0$ .

$2 x - 2 = 0$

Step 3.3.2

Solve $2 x - 2 = 0$ for $x$ .

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Step 3.3.2.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 3.3.2.2

Divide each term in $2 x = 2$ by $2$ and simplify.

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Step 3.3.2.2.1

Divide each term in $2 x = 2$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2}$

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

Step 3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{2}$

Step 3.3.2.2.3

Simplify the right side.

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Step 3.3.2.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 3.4

Set $\frac{1}{1 + {(x^{2} - 2 x)}^{2}}$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $\frac{1}{1 + {(x^{2} - 2 x)}^{2}}$ equal to $0$ .

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

Step 3.4.2

Solve $\frac{1}{1 + {(x^{2} - 2 x)}^{2}} = 0$ for $x$ .

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Step 3.4.2.1

Set the numerator equal to zero.

$1 = 0$

Step 3.4.2.2

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3.5

The final solution is all the values that make $(2 x - 2) (\frac{1}{1 + {(x^{2} - 2 x)}^{2}}) = 0$ true.

$x = 1$

Step 4

The values which make the derivative equal to $0$ are $1$ .

$1$

Step 5

After finding the point that makes the derivative $f' (x) = (2 x - 2) (\frac{1}{1 + {(x^{2} - 2 x)}^{2}})$ equal to $0$ or undefined, the interval to check where $f (x) = \arctan (x^{2} - 2 x)$ is increasing and where it is decreasing is $(- \infty, 1) \cup (1, \infty)$ .

$(- \infty, 1) \cup (1, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = (2 (0) - 2) (\frac{1}{1 + {({(0)}^{2} - 2 \cdot 0)}^{2}})$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = (2 (0) - 2) (\frac{1}{1 + {(0 - 2 \cdot 0)}^{2}})$

Step 6.2.1.1.2

Multiply $- 2$ by $0$ .

$f' (0) = (2 (0) - 2) (\frac{1}{1 + {(0 + 0)}^{2}})$

Step 6.2.1.2

Add $0$ and $0$ .

$f' (0) = (2 (0) - 2) (\frac{1}{1 + 0^{2}})$

Step 6.2.1.3

Raising $0$ to any positive power yields $0$ .

$f' (0) = (2 (0) - 2) (\frac{1}{1 + 0})$

Step 6.2.1.4

Add $1$ and $0$ .

$f' (0) = (2 (0) - 2) (\frac{1}{1})$

Step 6.2.2

Reduce the expression by cancelling the common factors.

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Step 6.2.2.1

Cancel the common factor of $1$ .

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Step 6.2.2.1.1

Cancel the common factor.

$f' (0) = (2 (0) - 2) (\frac{1}{1})$

Step 6.2.2.1.2

Rewrite the expression.

$f' (0) = (2 (0) - 2) \cdot 1$

Step 6.2.2.2

Simplify the expression.

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Step 6.2.2.2.1

Multiply $2 (0) - 2$ by $1$ .

$f' (0) = 2 (0) - 2$

Step 6.2.2.2.2

Multiply $2$ by $0$ .

$f' (0) = 0 - 2$

Step 6.2.2.2.3

Subtract $2$ from $0$ .

$f' (0) = - 2$

Step 6.2.3

The final answer is $- 2$ .

$- 2$

Step 6.3

At $x = 0$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(- \infty, 1)$ .

Decreasing on $(- \infty, 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = (2 (2) - 2) (\frac{1}{1 + {({(2)}^{2} - 2 \cdot 2)}^{2}})$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

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

$f' (2) = (2 (2) - 2) (\frac{1}{1 + {(4 - 2 \cdot 2)}^{2}})$

Step 7.2.1.1.2

Multiply $- 2$ by $2$ .

$f' (2) = (2 (2) - 2) (\frac{1}{1 + {(4 - 4)}^{2}})$

Step 7.2.1.2

Subtract $4$ from $4$ .

$f' (2) = (2 (2) - 2) (\frac{1}{1 + 0^{2}})$

Step 7.2.1.3

Raising $0$ to any positive power yields $0$ .

$f' (2) = (2 (2) - 2) (\frac{1}{1 + 0})$

Step 7.2.1.4

Add $1$ and $0$ .

$f' (2) = (2 (2) - 2) (\frac{1}{1})$

Step 7.2.2

Reduce the expression by cancelling the common factors.

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Step 7.2.2.1

Cancel the common factor of $1$ .

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Step 7.2.2.1.1

Cancel the common factor.

$f' (2) = (2 (2) - 2) (\frac{1}{1})$

Step 7.2.2.1.2

Rewrite the expression.

$f' (2) = (2 (2) - 2) \cdot 1$

Step 7.2.2.2

Simplify the expression.

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Step 7.2.2.2.1

Multiply $2 (2) - 2$ by $1$ .

$f' (2) = 2 (2) - 2$

Step 7.2.2.2.2

Multiply $2$ by $2$ .

$f' (2) = 4 - 2$

Step 7.2.2.2.3

Subtract $2$ from $4$ .

$f' (2) = 2$

Step 7.2.3

The final answer is $2$ .

$2$

Step 7.3

At $x = 2$ the derivative is $2$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(1, \infty)$

Decreasing on: $(- \infty, 1)$

Step 9