Calculus Examples

$f (x) = x^{3} + 3 x + 5$

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [5]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [5]$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.2.2

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

$3 x^{2} + 3 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.2.3

Multiply $3$ by $1$ .

$3 x^{2} + 3 + \frac{d}{d x} [5]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 x^{2} + 3 + 0$

Step 1.1.3.2

Add $3 x^{2} + 3$ and $0$ .

$f' (x) = 3 x^{2} + 3$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 3$ .

$3 x^{2} + 3$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 3 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} + 3 = 0$

Step 2.2

Subtract $3$ from both sides of the equation.

$3 x^{2} = - 3$

Step 2.3

Divide each term in $3 x^{2} = - 3$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 x^{2} = - 3$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 3}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 3}{3}$

Step 2.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 3}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 3$ by $3$ .

$x^{2} = - 1$

Step 2.4

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

$x = \pm \sqrt{- 1}$

Step 2.5

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

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 $f' (x) = 3 x^{2} + 3$ equal to $0$ or undefined. The interval to check if $f (x) = x^{3} + 3 x + 5$ 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 $f' (x) = 3 x^{2} + 3$ 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.

$f' (1) = 3 {(1)}^{2} + 3$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

One to any power is one.

$f' (1) = 3 \cdot 1 + 3$

Step 5.2.1.2

Multiply $3$ by $1$ .

$f' (1) = 3 + 3$

Step 5.2.2

Add $3$ and $3$ .

$f' (1) = 6$

Step 5.2.3

The final answer is $6$ .

$6$

Step 6

The result of substituting $1$ into $f' (x) = 3 x^{2} + 3$ is $6$ , which is positive, so the graph is increasing on the interval $(- \infty, \infty)$ .

Increasing on $(- \infty, \infty)$ since $3 x^{2} + 3 > 0$

Step 7

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

Always Increasing

Step 8