Calculus Examples

$f (x) = \tan (\frac{π x}{2})$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 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 $f (x) = \tan (x)$ and $g (x) = \frac{π x}{2}$ .

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Step 1.1.1.1

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

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

Step 1.1.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [\frac{π x}{2}]$

Step 1.1.1.3

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

$\sec^{2} (\frac{π x}{2}) \frac{d}{d x} [\frac{π x}{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$\sec^{2} (\frac{π x}{2}) (\frac{π}{2} \frac{d}{d x} [x])$

Step 1.1.2.2

Combine $\frac{π}{2}$ and $\sec^{2} (\frac{π x}{2})$ .

$\frac{π \sec^{2} (\frac{π x}{2})}{2} \frac{d}{d x} [x]$

Step 1.1.2.3

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

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

Step 1.1.2.4

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

$f' (x) = \frac{π \sec^{2} (\frac{π x}{2})}{2}$

Step 1.2

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

$\frac{π \sec^{2} (\frac{π x}{2})}{2}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{π \sec^{2} (\frac{π x}{2})}{2} = 0$

Step 2.2

Set the numerator equal to zero.

$π \sec^{2} (\frac{π x}{2}) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $π \sec^{2} (\frac{π x}{2}) = 0$ by $π$ and simplify.

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Step 2.3.1.1

Divide each term in $π \sec^{2} (\frac{π x}{2}) = 0$ by $π$ .

$\frac{π \sec^{2} (\frac{π x}{2})}{π} = \frac{0}{π}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $π$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{π \sec^{2} (\frac{π x}{2})}{π} = \frac{0}{π}$

Step 2.3.1.2.1.2

Divide $\sec^{2} (\frac{π x}{2})$ by $1$ .

$\sec^{2} (\frac{π x}{2}) = \frac{0}{π}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $π$ .

$\sec^{2} (\frac{π x}{2}) = 0$

Step 2.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (\frac{π x}{2}) = \pm \sqrt{0}$

Step 2.3.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.3.1

Rewrite $0$ as $0^{2}$ .

$\sec (\frac{π x}{2}) = \pm \sqrt{0^{2}}$

Step 2.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\sec (\frac{π x}{2}) = \pm 0$

Step 2.3.3.3

Plus or minus $0$ is $0$ .

$\sec (\frac{π x}{2}) = 0$

Step 2.3.4

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the argument in $\sec (\frac{π x}{2})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{π x}{2} = \frac{π}{2} + π n$ , for any integer $n$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Multiply both sides of the equation by $\frac{2}{π}$ .

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2

Simplify both sides of the equation.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Simplify $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 3.2.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π x}{2} = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2

Cancel the common factor of $π$ .

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Step 3.2.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} (\frac{π}{2} + π n)$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $\frac{2}{π} (\frac{π}{2} + π n)$ .

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Step 3.2.2.2.1.1

Apply the distributive property.

$x = \frac{2}{π} \cdot \frac{π}{2} + \frac{2}{π} (π n)$

Step 3.2.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.2.1

Cancel the common factor.

$x = \frac{2}{π} \cdot \frac{π}{2} + \frac{2}{π} (π n)$

Step 3.2.2.2.1.2.2

Rewrite the expression.

$x = \frac{1}{π} π + \frac{2}{π} (π n)$

Step 3.2.2.2.1.3

Cancel the common factor of $π$ .

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Step 3.2.2.2.1.3.1

Cancel the common factor.

$x = \frac{1}{π} π + \frac{2}{π} (π n)$

Step 3.2.2.2.1.3.2

Rewrite the expression.

$x = 1 + \frac{2}{π} (π n)$

Step 3.2.2.2.1.4

Cancel the common factor of $π$ .

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Step 3.2.2.2.1.4.1

Factor $π$ out of $π n$ .

$x = 1 + \frac{2}{π} (π (n))$

Step 3.2.2.2.1.4.2

Cancel the common factor.

$x = 1 + \frac{2}{π} (π n)$

Step 3.2.2.2.1.4.3

Rewrite the expression.

$x = 1 + 2 n$

Step 3.2.3

Reorder $1$ and $2 n$ .

$x = 2 n + 1$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = 2 n + 1}$ $n$ , for any integer $n$

Step 4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found