Calculus Examples

$f (x) = \arccos (s) (x) - 3$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\arccos (s) (x) - 3$ with respect to $x$ is $\frac{d}{d x} [\arccos (s) (x)] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [\arccos (s) (x)] + \frac{d}{d x} [- 3]$

Step 1.2

Evaluate $\frac{d}{d x} [\arccos (s) (x)]$ .

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Step 1.2.1

Since $\arccos (s)$ is constant with respect to $x$ , the derivative of $\arccos (s) x$ with respect to $x$ is $\arccos (s) \frac{d}{d x} [x]$ .

$\arccos (s) \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 1.2.2

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

$\arccos (s) \cdot 1 + \frac{d}{d x} [- 3]$

Step 1.2.3

Multiply $\arccos (s)$ by $1$ .

$\arccos (s) + \frac{d}{d x} [- 3]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\arccos (s) + 0$

Step 1.3.2

Add $\arccos (s)$ and $0$ .

$\arccos (s)$

Step 2

Since $\arccos (s)$ is constant with respect to $x$ , the derivative of $\arccos (s)$ with respect to $x$ is $0$ .

$f'' (x) = 0$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\arccos (s) = 0$

Step 4

Take the inverse arccosine of both sides of the equation to extract $s$ from inside the arccosine.

$s = \cos (0)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\cos (0)$ is $1$ .

$s = 1$

Step 6

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$0$

Step 7

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 8