Calculus Examples

$x^{4} - 3 x^{2} + 2$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{4} - 3 x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [2]$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $- 3$ .

$4 x^{3} - 6 x + \frac{d}{d x} [2]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$4 x^{3} - 6 x + 0$

Step 1.3.2

Add $4 x^{3} - 6 x$ and $0$ .

$4 x^{3} - 6 x$

Step 2

Set the derivative equal to $0$ then solve the equation $4 x^{3} - 6 x = 0$ .

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Step 2.1

Factor $2 x$ out of $4 x^{3} - 6 x$ .

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Step 2.1.1

Factor $2 x$ out of $4 x^{3}$ .

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

Step 2.1.2

Factor $2 x$ out of $- 6 x$ .

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

Step 2.1.3

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

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

Step 2.2

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

$x = 0$

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

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

Add $3$ to both sides of the equation.

$2 x^{2} = 3$

Step 2.4.2.2

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

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Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

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

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

Step 2.4.2.3

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

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

Step 2.4.2.4

Simplify $\pm \sqrt{\frac{3}{2}}$ .

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Step 2.4.2.4.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

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

Step 2.4.2.4.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.4.2.4.3

Combine and simplify the denominator.

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Step 2.4.2.4.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.4.2.4.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4.2.4.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4.2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.4.2.4.3.5

Add $1$ and $1$ .

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

Step 2.4.2.4.3.6

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

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Step 2.4.2.4.3.6.1

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

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

Step 2.4.2.4.3.6.2

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

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

Step 2.4.2.4.3.6.3

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

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

Step 2.4.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.4.3.6.4.1

Cancel the common factor.

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

Step 2.4.2.4.3.6.4.2

Rewrite the expression.

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

Step 2.4.2.4.3.6.5

Evaluate the exponent.

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

Step 2.4.2.4.4

Simplify the numerator.

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Step 2.4.2.4.4.1

Combine using the product rule for radicals.

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

Step 2.4.2.4.4.2

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{6}}{2}$

Step 2.4.2.5

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

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Step 2.4.2.5.1

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

$x = \frac{\sqrt{6}}{2}$

Step 2.4.2.5.2

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

$x = - \frac{\sqrt{6}}{2}$

Step 2.4.2.5.3

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

$x = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 2.5

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

$x = 0, \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 3

Solve the original function $f (x) = x^{4} - 3 x^{2} + 2$ at $x = 0$ .

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Step 3.1

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

$f (0) = {(0)}^{4} - 3 {(0)}^{2} + 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

$f (0) = 0 - 3 \cdot 0 + 2$

Step 3.2.1.3

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 + 2$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 2$

Step 3.2.2.2

Add $0$ and $2$ .

$f (0) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 4

Solve the original function $f (x) = x^{4} - 3 x^{2} + 2$ at $x = \frac{\sqrt{6}}{2}$ .

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Step 4.1

Replace the variable $x$ with $\frac{\sqrt{6}}{2}$ in the expression.

$f (\frac{\sqrt{6}}{2}) = {(\frac{\sqrt{6}}{2})}^{4} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

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

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{{\sqrt{6}}^{4}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 4.2.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{{(6^{\frac{1}{2}})}^{4}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.2

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

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{1}{2} \cdot 4}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.3

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

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{4}{2}}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.4

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

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Step 4.2.1.2.1.4.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{2 \cdot 2}{2}}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.4.2

Cancel the common factors.

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Step 4.2.1.2.1.4.2.1

Factor $2$ out of $2$ .

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{2}) = \frac{6^{\frac{2}{1}}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.1.4.2.4

Divide $2$ by $1$ .

$f (\frac{\sqrt{6}}{2}) = \frac{6^{2}}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.2.2

Raise $6$ to the power of $2$ .

$f (\frac{\sqrt{6}}{2}) = \frac{36}{2^{4}} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.3

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

$f (\frac{\sqrt{6}}{2}) = \frac{36}{16} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.4

Cancel the common factor of $36$ and $16$ .

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Step 4.2.1.4.1

Factor $4$ out of $36$ .

$f (\frac{\sqrt{6}}{2}) = \frac{4 (9)}{16} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.4.2

Cancel the common factors.

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Step 4.2.1.4.2.1

Factor $4$ out of $16$ .

$f (\frac{\sqrt{6}}{2}) = \frac{4 \cdot 9}{4 \cdot 4} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{2}) = \frac{4 \cdot 9}{4 \cdot 4} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 {(\frac{\sqrt{6}}{2})}^{2} + 2$

Step 4.2.1.5

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{{\sqrt{6}}^{2}}{2^{2}} + 2$

Step 4.2.1.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.2.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{{(6^{\frac{1}{2}})}^{2}}{2^{2}} + 2$

Step 4.2.1.6.2

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

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{6^{\frac{1}{2} \cdot 2}}{2^{2}} + 2$

Step 4.2.1.6.3

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

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{6^{\frac{2}{2}}}{2^{2}} + 2$

Step 4.2.1.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.6.4.1

Cancel the common factor.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{6^{\frac{2}{2}}}{2^{2}} + 2$

Step 4.2.1.6.4.2

Rewrite the expression.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{6}{2^{2}} + 2$

Step 4.2.1.6.5

Evaluate the exponent.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{6}{2^{2}} + 2$

Step 4.2.1.7

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

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 (\frac{6}{4}) + 2$

Step 4.2.1.8

Cancel the common factor of $6$ and $4$ .

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Step 4.2.1.8.1

Factor $2$ out of $6$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{2 (3)}{4} + 2$

Step 4.2.1.8.2

Cancel the common factors.

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Step 4.2.1.8.2.1

Factor $2$ out of $4$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{2 \cdot 3}{2 \cdot 2} + 2$

Step 4.2.1.8.2.2

Cancel the common factor.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 \frac{2 \cdot 3}{2 \cdot 2} + 2$

Step 4.2.1.8.2.3

Rewrite the expression.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - 3 (\frac{3}{2}) + 2$

Step 4.2.1.9

Multiply $- 3 (\frac{3}{2})$ .

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Step 4.2.1.9.1

Combine $- 3$ and $\frac{3}{2}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} + \frac{- 3 \cdot 3}{2} + 2$

Step 4.2.1.9.2

Multiply $- 3$ by $3$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} + \frac{- 9}{2} + 2$

Step 4.2.1.10

Move the negative in front of the fraction.

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9}{2} + 2$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - (\frac{9}{2} \cdot \frac{2}{2}) + 2$

Step 4.2.2.2

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2} + 2$

Step 4.2.2.3

Write $2$ as a fraction with denominator $1$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2} + \frac{2}{1}$

Step 4.2.2.4

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2} + \frac{2}{1} \cdot \frac{4}{4}$

Step 4.2.2.5

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2} + \frac{2 \cdot 4}{4}$

Step 4.2.2.6

Multiply $2$ by $2$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{4} + \frac{2 \cdot 4}{4}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{\sqrt{6}}{2}) = \frac{9 - 9 \cdot 2 + 2 \cdot 4}{4}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 9$ by $2$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9 - 18 + 2 \cdot 4}{4}$

Step 4.2.4.2

Multiply $2$ by $4$ .

$f (\frac{\sqrt{6}}{2}) = \frac{9 - 18 + 8}{4}$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Subtract $18$ from $9$ .

$f (\frac{\sqrt{6}}{2}) = \frac{- 9 + 8}{4}$

Step 4.2.5.2

Add $- 9$ and $8$ .

$f (\frac{\sqrt{6}}{2}) = \frac{- 1}{4}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (\frac{\sqrt{6}}{2}) = - \frac{1}{4}$

Step 4.2.6

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Step 5

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

$y = 2, y = - \frac{1}{4}, y = - \frac{1}{4}$

Step 6