Calculus Examples

$\tan (x)$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

$\sec^{2} (x)$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

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

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

Step 2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.1

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

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

Step 2.3.2

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

$\sec (x) = \pm 0$

Step 2.3.3

Plus or minus $0$ is $0$ .

$\sec (x) = 0$

Step 2.4

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

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

Step 3.2

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 $0$ .

${x | x = \frac{π}{2} + π n}$ $n$ , 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