Calculus Examples

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

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

$3 x^{2}$

Set the derivative equal to $0$ then solve for $x$ .

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Rewrite $3 x^{2}$ as a set of linear factors.

$3 (x^{2}) = 0$

Divide each term by $3$ and simplify.

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Divide each term in $3 (x^{2}) = 0$ by $3$ .

$\frac{3 (x^{2})}{3} = \frac{0}{3}$

Simplify the left side of the $e q u a t i o n$ by cancelling the common factors.

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Multiply $3$ by $x^{2}$ to get $3 x^{2}$ .

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{0}{3}$

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

$x^{2} = \frac{0}{3}$

Divide $0$ by $3$ to get $0$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

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

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Simplify the right side of the equation.

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

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

$\pm 0$ is equal to $0$ .

$x = 0$

After finding the point that makes the derivative $f' (x) = 3 x^{2}$ equal to $0$ or undefined, the interval to check where $f (x) = x^{3}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

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

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

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

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

Simplify the result.

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Raise $- 1$ to the power of $2$ to get $1$ .

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

Multiply $3$ by $1$ to get $3$ .

$f' (- 1) = 3$

The final answer is $3$ .

$3$

At $x = - 1$ the derivative is $3$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

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

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

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

Simplify the result.

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Remove parentheses around $1$ .

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

One to any power is one.

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

Multiply $3$ by $1$ to get $3$ .

$f' (1) = 3$

The final answer is $3$ .

$3$

At $x = 1$ the derivative is $3$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

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

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

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