Calculus Examples

$f (x) = x^{2} - 2$

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

By the Sum Rule, the derivative of $x^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2]$ .

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

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 2]$

Step 1.1.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.4

Add $2 x$ and $0$ .

$f' (x) = 2 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x$ .

$2 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 x = 0$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 4

After finding the point that makes the derivative $f' (x) = 2 x$ equal to $0$ or undefined, the interval to check where $f (x) = x^{2} - 2$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

$(- \infty, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 2 (- 1)$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $- 1$ .

$f' (- 1) = - 2$

Step 5.2.2

The final answer is $- 2$ .

$- 2$

Step 5.3

At $x = - 1$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 2 (1)$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $1$ .

$f' (1) = 2$

Step 6.2.2

The final answer is $2$ .

$2$

Step 6.3

At $x = 1$ the derivative is $2$ . Since this is positive, the function is increasing on $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on: $(0, \infty)$

Decreasing on: $(- \infty, 0)$

Step 8