Calculus Examples

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

Step 1

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

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

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 $x^{4} - 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- 2$ by $1$ .

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

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 $0$ .

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

Step 2.3.2

Add $4 x^{3} - 2$ and $0$ .

$4 x^{3} - 2$

Step 3

Set the derivative equal to $0$ then solve the equation $4 x^{3} - 2 = 0$ .

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Step 3.1

Add $2$ to both sides of the equation.

$4 x^{3} = 2$

Step 3.2

Subtract $2$ from both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Factor $2$ out of $- 2$ .

$2 (2 x^{3}) + 2 (- 1) = 0$

Step 3.3.3

Factor $2$ out of $2 (2 x^{3}) + 2 (- 1)$ .

$2 (2 x^{3} - 1) = 0$

Step 3.4

Divide each term in $2 (2 x^{3} - 1) = 0$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 (2 x^{3} - 1) = 0$ by $2$ .

$\frac{2 (2 x^{3} - 1)}{2} = \frac{0}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 (2 x^{3} - 1)}{2} = \frac{0}{2}$

Step 3.4.2.1.2

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

$2 x^{3} - 1 = \frac{0}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $2$ .

$2 x^{3} - 1 = 0$

Step 3.5

Add $1$ to both sides of the equation.

$2 x^{3} = 1$

Step 3.6

Divide each term in $2 x^{3} = 1$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 x^{3} = 1$ by $2$ .

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

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

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

Step 3.6.2.1.2

Divide $x^{3}$ by $1$ .

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

Step 3.7

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

$x = \sqrt[3]{\frac{1}{2}}$

Step 3.8

Simplify $\sqrt[3]{\frac{1}{2}}$ .

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Step 3.8.1

Rewrite $\sqrt[3]{\frac{1}{2}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2}}$ .

$x = \frac{\sqrt[3]{1}}{\sqrt[3]{2}}$

Step 3.8.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[3]{2}}$

Step 3.8.3

Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 3.8.4

Combine and simplify the denominator.

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Step 3.8.4.1

Multiply $\frac{1}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.8.4.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 3.8.4.3

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

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

Step 3.8.4.4

Add $1$ and $2$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.8.4.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.8.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

Step 3.8.4.5.2

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

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

Step 3.8.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.8.4.5.4

Cancel the common factor of $3$ .

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Step 3.8.4.5.4.1

Cancel the common factor.

$x = \frac{{\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.8.4.5.4.2

Rewrite the expression.

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

Step 3.8.4.5.5

Evaluate the exponent.

$x = \frac{{\sqrt[3]{2}}^{2}}{2}$

Step 3.8.5

Simplify the numerator.

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Step 3.8.5.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$x = \frac{\sqrt[3]{2^{2}}}{2}$

Step 3.8.5.2

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

$x = \frac{\sqrt[3]{4}}{2}$

Step 4

Solve the original function $f (x) = x^{4} - 2 x + 1$ at $x = \frac{\sqrt[3]{4}}{2}$ .

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Step 4.1

Replace the variable $x$ with $\frac{\sqrt[3]{4}}{2}$ in the expression.

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

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

Apply the product rule to $\frac{\sqrt[3]{4}}{2}$ .

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

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Rewrite ${\sqrt[3]{4}}^{4}$ as $\sqrt[3]{4^{4}}$ .

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

Step 4.2.1.2.2

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

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{256}}{2^{4}} - 2 \frac{\sqrt[3]{4}}{2} + 1$

Step 4.2.1.2.3

Rewrite $256$ as $4^{3} \cdot 4$ .

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Step 4.2.1.2.3.1

Factor $64$ out of $256$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{\sqrt[3]{64 (4)}}{2^{4}} - 2 \frac{\sqrt[3]{4}}{2} + 1$

Step 4.2.1.2.3.2

Rewrite $64$ as $4^{3}$ .

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

Step 4.2.1.2.4

Pull terms out from under the radical.

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

Step 4.2.1.3

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

$f (\frac{\sqrt[3]{4}}{2}) = \frac{4 \sqrt[3]{4}}{16} - 2 \frac{\sqrt[3]{4}}{2} + 1$

Step 4.2.1.4

Cancel the common factor of $4$ and $16$ .

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Step 4.2.1.4.1

Factor $4$ out of $4 \sqrt[3]{4}$ .

$f (\frac{\sqrt[3]{4}}{2}) = \frac{4 (\sqrt[3]{4})}{16} - 2 \frac{\sqrt[3]{4}}{2} + 1$

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 $16$ .

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

Step 4.2.1.4.2.2

Cancel the common factor.

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

Step 4.2.1.4.2.3

Rewrite the expression.

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

Step 4.2.1.5

Cancel the common factor of $2$ .

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Step 4.2.1.5.1

Factor $2$ out of $- 2$ .

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

Step 4.2.1.5.2

Cancel the common factor.

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

Step 4.2.1.5.3

Rewrite the expression.

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

Step 4.2.1.6

Rewrite $- 1 \sqrt[3]{4}$ as $- \sqrt[3]{4}$ .

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

Step 4.2.2

To write $- \sqrt[3]{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 4.2.3

Combine fractions.

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Step 4.2.3.1

Combine $- \sqrt[3]{4}$ and $\frac{4}{4}$ .

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

Step 4.2.3.2

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

Multiply $4$ by $- 1$ .

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

Step 4.2.4.1.2

Subtract $4 \sqrt[3]{4}$ from $\sqrt[3]{4}$ .

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

Step 4.2.4.2

Move the negative in front of the fraction.

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

Step 4.2.5

The final answer is $- \frac{3 \sqrt[3]{4}}{4} + 1$ .

$- \frac{3 \sqrt[3]{4}}{4} + 1$

Step 5

The horizontal tangent line on function $f (x) = x^{4} - 2 x + 1$ is $y = - \frac{3 \sqrt[3]{4}}{4} + 1$ .

$y = - \frac{3 \sqrt[3]{4}}{4} + 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{3 \sqrt[3]{4}}{4} + 1$

Decimal Form:

$y = - 0.19055078 \dots$

Step 7