Calculus Examples

$y = x^{4} - 8 x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{4} - 8 x^{2})$

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 $x^{4} - 8 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{2}]$

Step 3.1.2

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

$4 x^{3} + \frac{d}{d x} [- 8 x^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [- 8 x^{2}]$ .

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Step 3.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 8 \frac{d}{d x} [x^{2}]$

Step 3.2.2

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

$4 x^{3} - 8 (2 x)$

Step 3.2.3

Multiply $2$ by $- 8$ .

$4 x^{3} - 16 x$

Step 4

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

$y' = 4 x^{3} - 16 x$

Step 5

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

$\frac{d y}{d x} = 4 x^{3} - 16 x$

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 the left side of the equation.

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Step 6.1.1

Factor $4 x$ out of $4 x^{3} - 16 x$ .

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Step 6.1.1.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 16 x = 0$

Step 6.1.1.2

Factor $4 x$ out of $- 16 x$ .

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

Step 6.1.1.3

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

$4 x (x^{2} - 4) = 0$

Step 6.1.2

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

$4 x (x^{2} - 2^{2}) = 0$

Step 6.1.3

Factor.

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Step 6.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$4 x ((x + 2) (x - 2)) = 0$

Step 6.1.3.2

Remove unnecessary parentheses.

$4 x (x + 2) (x - 2) = 0$

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 $0$ .

$x = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 6.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.6

The final solution is all the values that make $4 x (x + 2) (x - 2) = 0$ true.

$x = 0, - 2, 2$

Step 7

Solve for $y$ .

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Step 7.1

Remove parentheses.

$y = 0^{4} - 8 {(0)}^{2}$

Step 7.2

Remove parentheses.

$y = 0^{4} - 8 \cdot 0^{2}$

Step 7.3

Simplify $0^{4} - 8 \cdot 0^{2}$ .

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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 $0$ .

$y = 0 - 8 \cdot 0^{2}$

Step 7.3.1.2

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

$y = 0 - 8 \cdot 0$

Step 7.3.1.3

Multiply $- 8$ by $0$ .

$y = 0 + 0$

Step 7.3.2

Add $0$ and $0$ .

$y = 0$

Step 8

Simplify ${(- 2)}^{4} - 8 {(- 2)}^{2}$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Raise $- 2$ to the power of $4$ .

$y = 16 - 8 {(- 2)}^{2}$

Step 8.1.2

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

$y = 16 - 8 \cdot 4$

Step 8.1.3

Multiply $- 8$ by $4$ .

$y = 16 - 32$

Step 8.2

Subtract $32$ from $16$ .

$y = - 16$

Step 9

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

$(0, 0)$

$(- 2, - 16)$

$(2, - 16)$

Step 10

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