Calculus Examples

$y = 2 + 48 x^{2} + 32 x^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (2 + 48 x^{2} + 32 x^{4})$

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

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $2 + 48 x^{2} + 32 x^{4}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [32 x^{4}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [32 x^{4}]$

Step 3.1.2

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

$0 + \frac{d}{d x} [48 x^{2}] + \frac{d}{d x} [32 x^{4}]$

Step 3.2

Evaluate $\frac{d}{d x} [48 x^{2}]$ .

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Step 3.2.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x^{2}$ with respect to $x$ is $48 \frac{d}{d x} [x^{2}]$ .

$0 + 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [32 x^{4}]$

Step 3.2.2

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

$0 + 48 (2 x) + \frac{d}{d x} [32 x^{4}]$

Step 3.2.3

Multiply $2$ by $48$ .

$0 + 96 x + \frac{d}{d x} [32 x^{4}]$

Step 3.3

Evaluate $\frac{d}{d x} [32 x^{4}]$ .

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Step 3.3.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{4}$ with respect to $x$ is $32 \frac{d}{d x} [x^{4}]$ .

$0 + 96 x + 32 \frac{d}{d x} [x^{4}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 + 96 x + 32 (4 x^{3})$

Step 3.3.3

Multiply $4$ by $32$ .

$0 + 96 x + 128 x^{3}$

Step 3.4

Simplify.

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Step 3.4.1

Add $0$ and $96 x$ .

$96 x + 128 x^{3}$

Step 3.4.2

Reorder terms.

$128 x^{3} + 96 x$

Step 4

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

$y' = 128 x^{3} + 96 x$

Step 5

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

$\frac{d y}{d x} = 128 x^{3} + 96 x$

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 $32 x$ out of $128 x^{3} + 96 x$ .

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Step 6.1.1

Factor $32 x$ out of $128 x^{3}$ .

$32 x (4 x^{2}) + 96 x = 0$

Step 6.1.2

Factor $32 x$ out of $96 x$ .

$32 x (4 x^{2}) + 32 x (3) = 0$

Step 6.1.3

Factor $32 x$ out of $32 x (4 x^{2}) + 32 x (3)$ .

$32 x (4 x^{2} + 3) = 0$

Step 6.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x^{2} + 3 = 0$

Step 6.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.4

Set $4 x^{2} + 3$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $4 x^{2} + 3$ equal to $0$ .

$4 x^{2} + 3 = 0$

Step 6.4.2

Solve $4 x^{2} + 3 = 0$ for $x$ .

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Step 6.4.2.1

Subtract $3$ from both sides of the equation.

$4 x^{2} = - 3$

Step 6.4.2.2

Divide each term in $4 x^{2} = - 3$ by $4$ and simplify.

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Step 6.4.2.2.1

Divide each term in $4 x^{2} = - 3$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{- 3}{4}$

Step 6.4.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{- 3}{4}$

Step 6.4.2.2.2.1.2

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

$x^{2} = \frac{- 3}{4}$

Step 6.4.2.2.3

Simplify the right side.

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Step 6.4.2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{3}{4}$

Step 6.4.2.3

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

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

Step 6.4.2.4

Simplify $\pm \sqrt{- \frac{3}{4}}$ .

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Step 6.4.2.4.1

Rewrite $- \frac{3}{4}$ as ${(\frac{i}{2})}^{2} \cdot 3$ .

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Step 6.4.2.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{3}{4}}$

Step 6.4.2.4.1.2

Factor the perfect power $1^{2}$ out of $3$ .

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

Step 6.4.2.4.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{2^{2} \cdot 1}}$

Step 6.4.2.4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 3}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 3)}$

Step 6.4.2.4.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 3}$

Step 6.4.2.4.2

Pull terms out from under the radical.

$x = \pm \frac{i}{2} \sqrt{3}$

Step 6.4.2.4.3

Combine $\frac{i}{2}$ and $\sqrt{3}$ .

$x = \pm \frac{i \sqrt{3}}{2}$

Step 6.4.2.5

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

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Step 6.4.2.5.1

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

$x = \frac{i \sqrt{3}}{2}$

Step 6.4.2.5.2

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

$x = - \frac{i \sqrt{3}}{2}$

Step 6.4.2.5.3

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

$x = \frac{i \sqrt{3}}{2}, - \frac{i \sqrt{3}}{2}$

Step 6.5

The final solution is all the values that make $32 x (4 x^{2} + 3) = 0$ true.

$x = 0, \frac{i \sqrt{3}}{2}, - \frac{i \sqrt{3}}{2}$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = 2 + 48 \cdot 0^{2} + 32 {(0)}^{4}$

Step 7.2

Remove parentheses.

$y = 2 + 48 \cdot 0^{2} + 32 \cdot 0^{4}$

Step 7.3

Simplify $2 + 48 \cdot 0^{2} + 32 \cdot 0^{4}$ .

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = 2 + 48 \cdot 0 + 32 \cdot 0^{4}$

Step 7.3.1.2

Multiply $48$ by $0$ .

$y = 2 + 0 + 32 \cdot 0^{4}$

Step 7.3.1.3

Raising $0$ to any positive power yields $0$ .

$y = 2 + 0 + 32 \cdot 0$

Step 7.3.1.4

Multiply $32$ by $0$ .

$y = 2 + 0 + 0$

Step 7.3.2

Simplify by adding numbers.

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Step 7.3.2.1

Add $2$ and $0$ .

$y = 2 + 0$

Step 7.3.2.2

Add $2$ and $0$ .

$y = 2$

Step 8

Calculated $x$ values cannot contain imaginary components.

$\frac{i \sqrt{3}}{2}$ is not a valid value for x

Step 9

Calculated $x$ values cannot contain imaginary components.

$- \frac{i \sqrt{3}}{2}$ is not a valid value for x

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 2)$

Step 11