Calculus Examples

$f (x) = \tan^{-1} (x^{2})$ on $- 2$ , $2$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan^{-1} (x)$ and $g (x) = x^{2}$ .

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.1.3

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

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

Step 1.1.1.2

Differentiate using the Power Rule.

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Step 1.1.1.2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.1.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.1.2.1.2

Multiply $2$ by $2$ .

$\frac{1}{1 + x^{4}} \frac{d}{d x} [x^{2}]$

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

Combine fractions.

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Step 1.1.1.2.3.1

Combine $2$ and $\frac{1}{1 + x^{4}}$ .

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

Step 1.1.1.2.3.2

Combine $\frac{2}{1 + x^{4}}$ and $x$ .

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

Step 1.1.1.2.3.3

Reorder terms.

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x}{x^{4} + 1}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{2 x}{x^{4} + 1} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{2 x}{x^{4} + 1} = 0$

Step 1.2.2

Set the numerator equal to zero.

$2 x = 0$

Step 1.2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $\tan^{-1} (x^{2})$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\tan^{-1} ({(0)}^{2})$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

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

$\tan^{-1} (0)$

Step 1.4.1.2.2

The exact value of $\tan^{-1} (0)$ is $0$ .

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 2$ .

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Step 2.1.1

Substitute $- 2$ for $x$ .

$\tan^{-1} ({(- 2)}^{2})$

Step 2.1.2

Simplify.

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Step 2.1.2.1

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

$\tan^{-1} (4)$

Step 2.1.2.2

Evaluate $\tan^{-1} (4)$ .

$1.32581766$

Step 2.2

Evaluate at $x = 2$ .

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Step 2.2.1

Substitute $2$ for $x$ .

$\tan^{-1} ({(2)}^{2})$

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

$\tan^{-1} (4)$

Step 2.2.2.2

Evaluate $\tan^{-1} (4)$ .

$1.32581766$

Step 2.3

List all of the points.

$(- 2, 1.32581766), (2, 1.32581766)$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(- 2, 1.32581766), (2, 1.32581766)$

Absolute Minimum: $(0, 0)$

Step 4