Calculus Examples

$y = \tan (x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = \tan (x)$

Step 2

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

$\sec^{2} (x)$

Step 3

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

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Step 3.1

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

$\sec (x) = \pm \sqrt{0}$

Step 3.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

$\sec (x) = \pm 0$

Step 3.2.3

Plus or minus $0$ is $0$ .

$\sec (x) = 0$

Step 3.3

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 4

There are no solution found by setting the derivative equal to $0$ , $\sec^{2} (x) = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5