Calculus Examples

$y = x^{4} - 4 x^{2} + 1$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{4} - 4 x^{2} + 1$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [1]$

Step 2.1.2

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

$4 x^{3} + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [1]$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.2.2

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

$4 x^{3} - 4 (2 x) + \frac{d}{d x} [1]$

Step 2.2.3

Multiply $2$ by $- 4$ .

$4 x^{3} - 8 x + \frac{d}{d x} [1]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 x^{3} - 8 x + 0$

Step 2.3.2

Add $4 x^{3} - 8 x$ and $0$ .

$4 x^{3} - 8 x$

Step 3

Set the derivative equal to $0$ then solve the equation $4 x^{3} - 8 x = 0$ .

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Step 3.1

Factor $4 x$ out of $4 x^{3} - 8 x$ .

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Step 3.1.1

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

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

Step 3.1.2

Factor $4 x$ out of $- 8 x$ .

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

Step 3.1.3

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

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

Step 3.2

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

$x = 0$

$x^{2} - 2 = 0$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.4

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

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Step 3.4.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 3.4.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 3.4.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 3.4.2.2

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

$x = \pm \sqrt{2}$

Step 3.4.2.3

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

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Step 3.4.2.3.1

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

$x = \sqrt{2}$

Step 3.4.2.3.2

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

$x = - \sqrt{2}$

Step 3.4.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 3.5

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

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 4

Solve the original function $f (x) = x^{4} - 4 x^{2} + 1$ at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4} - 4 {(0)}^{2} + 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f (0) = 0 - 4 {(0)}^{2} + 1$

Step 4.2.1.2

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

$f (0) = 0 - 4 \cdot 0 + 1$

Step 4.2.1.3

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0 + 1$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 1$

Step 4.2.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 5

Solve the original function $f (x) = x^{4} - 4 x^{2} + 1$ at $x = \sqrt{2}$ .

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Step 5.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

$f (\sqrt{2}) = {(\sqrt{2})}^{4} - 4 {(\sqrt{2})}^{2} + 1$

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

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

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Step 5.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\sqrt{2}) = {(2^{\frac{1}{2}})}^{4} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{2}) = 2^{\frac{1}{2} \cdot 4} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$f (\sqrt{2}) = 2^{\frac{4}{2}} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 5.2.1.1.4.1

Factor $2$ out of $4$ .

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2}} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.4.2

Cancel the common factors.

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Step 5.2.1.1.4.2.1

Factor $2$ out of $2$ .

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2 (1)}} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.4.2.2

Cancel the common factor.

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2 \cdot 1}} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.4.2.3

Rewrite the expression.

$f (\sqrt{2}) = 2^{\frac{2}{1}} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.1.4.2.4

Divide $2$ by $1$ .

$f (\sqrt{2}) = 2^{2} - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.2

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

$f (\sqrt{2}) = 4 - 4 {(\sqrt{2})}^{2} + 1$

Step 5.2.1.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\sqrt{2}) = 4 - 4 {(2^{\frac{1}{2}})}^{2} + 1$

Step 5.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{2}) = 4 - 4 \cdot 2^{\frac{1}{2} \cdot 2} + 1$

Step 5.2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$f (\sqrt{2}) = 4 - 4 \cdot 2^{\frac{2}{2}} + 1$

Step 5.2.1.3.4

Cancel the common factor of $2$ .

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Step 5.2.1.3.4.1

Cancel the common factor.

$f (\sqrt{2}) = 4 - 4 \cdot 2^{\frac{2}{2}} + 1$

Step 5.2.1.3.4.2

Rewrite the expression.

$f (\sqrt{2}) = 4 - 4 \cdot 2 + 1$

Step 5.2.1.3.5

Evaluate the exponent.

$f (\sqrt{2}) = 4 - 4 \cdot 2 + 1$

Step 5.2.1.4

Multiply $- 4$ by $2$ .

$f (\sqrt{2}) = 4 - 8 + 1$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $8$ from $4$ .

$f (\sqrt{2}) = - 4 + 1$

Step 5.2.2.2

Add $- 4$ and $1$ .

$f (\sqrt{2}) = - 3$

Step 5.2.3

The final answer is $- 3$ .

$- 3$

Step 6

The horizontal tangent lines on function $f (x) = x^{4} - 4 x^{2} + 1$ are $y = 1, y = - 3, y = - 3$ .

$y = 1, y = - 3, y = - 3$

Step 7