Calculus Examples

$y = 9 x - 3 x^{2} + x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (9 x - 3 x^{2} + x^{3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $9 x - 3 x^{2} + x^{3}$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [x^{3}]$ .

$\frac{d}{d x} [9 x] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [x^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 3.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$9 \frac{d}{d x} [x] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [x^{3}]$

Step 3.2.2

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

$9 \cdot 1 + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [x^{3}]$

Step 3.2.3

Multiply $9$ by $1$ .

$9 + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [x^{3}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 3.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$9 - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x^{3}]$

Step 3.3.2

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

$9 - 3 (2 x) + \frac{d}{d x} [x^{3}]$

Step 3.3.3

Multiply $2$ by $- 3$ .

$9 - 6 x + \frac{d}{d x} [x^{3}]$

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

$9 - 6 x + 3 x^{2}$

Step 3.4.2

Reorder terms.

$3 x^{2} - 6 x + 9$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 3 x^{2} - 6 x + 9$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 3 x^{2} - 6 x + 9$

Step 6

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 6.1

Factor $3$ out of $3 x^{2} - 6 x + 9$ .

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Step 6.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 6 x + 9 = 0$

Step 6.1.2

Factor $3$ out of $- 6 x$ .

$3 (x^{2}) + 3 (- 2 x) + 9 = 0$

Step 6.1.3

Factor $3$ out of $9$ .

$3 x^{2} + 3 (- 2 x) + 3 \cdot 3 = 0$

Step 6.1.4

Factor $3$ out of $3 x^{2} + 3 (- 2 x)$ .

$3 (x^{2} - 2 x) + 3 \cdot 3 = 0$

Step 6.1.5

Factor $3$ out of $3 (x^{2} - 2 x) + 3 \cdot 3$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $x^{2} - 2 x + 3$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $3$ .

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

Step 6.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.4

Substitute the values $a = 1$ , $b = - 2$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 6.5

Simplify.

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 6.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 6.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 6.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 6.5.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 6.5.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 6.5.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.5.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.5.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 6.5.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 6.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 6.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2}$

Step 6.5.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 6.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.6.1

Simplify the numerator.

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Step 6.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 6.6.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 6.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 6.6.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 6.6.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 6.6.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 6.6.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.6.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.6.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.6.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 6.6.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 6.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 6.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2}$

Step 6.6.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 6.6.4

Change the $\pm$ to $+$ .

$x = 1 + i \sqrt{2}$

Step 6.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.7.1

Simplify the numerator.

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Step 6.7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 6.7.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 6.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 6.7.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 6.7.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 6.7.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 6.7.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.7.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.7.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.7.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.7.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 6.7.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 6.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 6.7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{2}}{2}$

Step 6.7.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 6.7.4

Change the $\pm$ to $-$ .

$x = 1 - i \sqrt{2}$

Step 6.8

The final answer is the combination of both solutions.

$x = 1 + i \sqrt{2}, 1 - i \sqrt{2}$

Step 7

Calculated $x$ values cannot contain imaginary components.

$1 + i \sqrt{2}$ is not a valid value for x

Step 8

Calculated $x$ values cannot contain imaginary components.

$1 - i \sqrt{2}$ is not a valid value for x

Step 9

No points that set $\frac{d y}{d x} = 0$ are on the real number plane.

No Points

Step 10