Calculus Examples

$f (x) = x - 1$

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 $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.2

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

$1 + \frac{d}{d x} [- 1]$

Step 1.1.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.4

Add $1$ and $0$ .

$f' (x) = 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $1$ .

$1$

Step 2

Set the first derivative equal to $0$ then solve the equation $1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$1 = 0$

Step 2.2

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

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

No critical points found

Step 4

No points make the derivative $f' (x) = 1$ equal to $0$ or undefined. The interval to check if $f (x) = x - 1$ is increasing or decreasing is $(- \infty, \infty)$ .

$(- \infty, \infty)$

Step 5

Substitute any number, such as $1$ , from the interval $(- \infty, \infty)$ in the derivative $f' (x) = 1$ to check if the result is negative or positive. If the result is negative, the graph is decreasing on the interval $(- \infty, \infty)$ . If the result is positive, the graph is increasing on the interval $(- \infty, \infty)$ .

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Step 5.1

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

$f' (1) = 1$

Step 5.2

The final answer is $1$ .

$1$

Step 6

The result of substituting $1$ into $f' (x) = 1$ is $1$ , which is positive, so the graph is increasing on the interval $(- \infty, \infty)$ .

Increasing on $(- \infty, \infty)$ since $1 > 0$

Step 7

Increasing over the interval $(- \infty, \infty)$ means that the function is always increasing.

Always Increasing

Step 8