Calculus Examples

$y (x) = x^{4} - 4 x + 4$

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 4$ by $1$ .

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

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 1.3.2

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

$4 x^{3} - 4$

Step 2

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

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Step 2.1

Add $4$ to both sides of the equation.

$4 x^{3} = 4$

Step 2.2

Subtract $4$ from both sides of the equation.

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

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

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

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Step 2.3.1.1

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

$4 (x^{3}) - 4 = 0$

Step 2.3.1.2

Factor $4$ out of $- 4$ .

$4 (x^{3}) + 4 (- 1) = 0$

Step 2.3.1.3

Factor $4$ out of $4 (x^{3}) + 4 (- 1)$ .

$4 (x^{3} - 1) = 0$

Step 2.3.2

Rewrite $1$ as $1^{3}$ .

$4 (x^{3} - 1^{3}) = 0$

Step 2.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$4 ((x - 1) (x^{2} + x \cdot 1 + 1^{2})) = 0$

Step 2.3.4

Factor.

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Step 2.3.4.1

Simplify.

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Step 2.3.4.1.1

Multiply $x$ by $1$ .

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

Step 2.3.4.1.2

One to any power is one.

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

Step 2.3.4.2

Remove unnecessary parentheses.

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

Step 2.4

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

$x - 1 = 0$

$x^{2} + x + 1 = 0$

Step 2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

Set $x^{2} + x + 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x^{2} + x + 1$ equal to $0$ .

$x^{2} + x + 1 = 0$

Step 2.6.2

Solve $x^{2} + x + 1 = 0$ for $x$ .

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Step 2.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.6.2.3

Simplify.

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Step 2.6.2.3.1

Simplify the numerator.

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Step 2.6.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.6.2.3.1.3

Subtract $4$ from $1$ .

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

Step 2.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 2.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.6.2.3.2

Multiply $2$ by $1$ .

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

Step 2.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.2.4.1

Simplify the numerator.

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Step 2.6.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.6.2.4.1.3

Subtract $4$ from $1$ .

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

Step 2.6.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.6.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 2.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.6.2.4.2

Multiply $2$ by $1$ .

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

Step 2.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{2}$

Step 2.6.2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{2}$

Step 2.6.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

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

Step 2.6.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{- 1 (1 - i \sqrt{3})}{2}$

Step 2.6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{2}$

Step 2.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.6.2.5.1

Simplify the numerator.

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Step 2.6.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.6.2.5.1.3

Subtract $4$ from $1$ .

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

Step 2.6.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.6.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 2.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.6.2.5.2

Multiply $2$ by $1$ .

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

Step 2.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{2}$

Step 2.6.2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.6.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

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

Step 2.6.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{- 1 (1 + i \sqrt{3})}{2}$

Step 2.6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{2}$

Step 2.6.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 2.7

The final solution is all the values that make $4 (x - 1) (x^{2} + x + 1) = 0$ true.

$x = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 3

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

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

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

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

One to any power is one.

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

Step 3.2.1.2

Multiply $- 4$ by $1$ .

$f (1) = 1 - 4 + 4$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $4$ from $1$ .

$f (1) = - 3 + 4$

Step 3.2.2.2

Add $- 3$ and $4$ .

$f (1) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

A tangent line cannot be found at an imaginary point. The point at $x = 1$ does not exist on the real coordinate system.

A tangent cannot be found from the root $1$

Step 5

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

$y = 1$

Step 6