Calculus Examples

$y = | x^{2} - x |$

Differentiate both sides of the equation.

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

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - x$ .

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To apply the Chain Rule, set $u$ as $x^{2} - x$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - x]$

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - x]$

Replace all occurrences of $u$ with $x^{2} - x$ .

$\frac{x^{2} - x}{| x^{2} - x |} \frac{d}{d x} [x^{2} - x]$

Differentiate.

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

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

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

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

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{x^{2} - x}{| x^{2} - x |} (2 x - \frac{d}{d x} [x])$

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

$\frac{x^{2} - x}{| x^{2} - x |} (2 x - 1 \cdot 1)$

Simplify the expression.

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Multiply $- 1$ by $1$ .

$\frac{x^{2} - x}{| x^{2} - x |} (2 x - 1)$

Reorder terms.

$(2 x - 1) \frac{x^{2} - x}{| x^{2} - x |}$

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

$y' = (2 x - 1) (\frac{x^{2} - x}{| x^{2} - x |})$

Simplify.

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

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

$y' = (2 x - 1) (\frac{x \cdot x - x}{| x^{2} - x |})$

Factor $x$ out of $- x$ .

$y' = (2 x - 1) (\frac{x \cdot x + x \cdot - 1}{| x^{2} - x |})$

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$y' = (2 x - 1) (\frac{x (x - 1)}{| x^{2} - x |})$

Multiply $2 x - 1$ and $\frac{x (x - 1)}{| x^{2} - x |}$ .

$y' = \frac{(2 x - 1) (x (x - 1))}{| x^{2} - x |}$

Reorder factors in $\frac{(2 x - 1) x (x - 1)}{| x^{2} - x |}$ .

$y' = \frac{x (2 x - 1) (x - 1)}{| x^{2} - x |}$

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

$\frac{d y}{d x} = \frac{x (2 x - 1) (x - 1)}{| x^{2} - x |}$

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Multiply each term by $| x^{2} - x |$ and simplify.

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Multiply each term in $\frac{x (2 x - 1) (x - 1)}{| x^{2} - x |} = 0$ by $| x^{2} - x |$ .

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

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $| x^{2} - x |$ .

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Write $| x^{2} - x |$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $| x^{2} - x |$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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Multiply $\frac{x (2 x - 1) (x - 1)}{1}$ and $\frac{1}{1}$ .

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

Divide $x (2 x - 1) (x - 1)$ by $1$ .

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

Simplify terms.

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Apply the distributive property.

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

Reorder.

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Rewrite using the commutative property of multiplication.

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

Move $- 1$ to the left of $x$ .

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

Simplify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Expand $(2 x^{2} - x) (x - 1)$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$2 x^{2} x + 2 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 = 0 \cdot | x^{2} - x |$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x^{2}$ by $x$ by adding the exponents.

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Move $x$ .

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

Multiply $x$ by $x^{2}$ .

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Raise $x$ to the power of $1$ .

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

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

$2 x^{1 + 2} + 2 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 = 0 \cdot | x^{2} - x |$

Add $1$ and $2$ .

$2 x^{3} + 2 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 = 0 \cdot | x^{2} - x |$

Multiply $- 1$ by $2$ .

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

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- x \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$2 x^{3} - 2 x^{2} - x^{2} + 1 x = 0 \cdot | x^{2} - x |$

Multiply $x$ by $1$ .

$2 x^{3} - 2 x^{2} - x^{2} + x = 0 \cdot | x^{2} - x |$

Subtract $x^{2}$ from $- 2 x^{2}$ .

$2 x^{3} - 3 x^{2} + x = 0 \cdot | x^{2} - x |$

Multiply $0$ by $| x^{2} - x |$ .

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

Factor the left side of the equation.

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

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

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

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

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

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

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

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

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

Factor.

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Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Factor $- 3$ out of $- 3 x$ .

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

Rewrite $- 3$ as $- 2$ plus $- 1$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

$x (2 x (x - 1) - 1 (x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$x ((x - 1) (2 x - 1)) = 0$

Remove unnecessary parentheses.

$x (x - 1) (2 x - 1) = 0$

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

$x = 0$

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

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Set the factor equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Set the factor equal to $0$ .

$2 x - 1 = 0$

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = 1$ by $2$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

The solution is the result of $x = 0, x - 1 = 0,$ and $2 x - 1 = 0$ .

$x = 0, 1, \frac{1}{2}$

Exclude the solutions that do not make $\frac{x (2 x - 1) (x - 1)}{| x^{2} - x |} = 0$ true.

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

Simplify $| {(\frac{1}{2})}^{2} - (\frac{1}{2}) |$ .

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Simplify each term.

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Apply the product rule to $\frac{1}{2}$ .

$y = | \frac{1^{2}}{2^{2}} - (\frac{1}{2}) |$

One to any power is one.

$y = | \frac{1}{2^{2}} - (\frac{1}{2}) |$

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

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

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = | \frac{1}{4} - \frac{1}{2} \cdot \frac{2}{2} |$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$y = | \frac{1}{4} - \frac{1 \cdot 2}{2 \cdot 2} |$

Multiply $2$ by $2$ .

$y = | \frac{1}{4} - \frac{1 \cdot 2}{4} |$

Combine the numerators over the common denominator.

$y = | \frac{1 - 1 \cdot 2}{4} |$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

$y = | \frac{- 1}{4} |$

Move the negative in front of the fraction.

$y = | - \frac{1}{4} |$

$- \frac{1}{4}$ is approximately $- 0.25$ which is negative so negate $- \frac{1}{4}$ and remove the absolute value

$y = \frac{1}{4}$

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

$(\frac{1}{2}, \frac{1}{4})$

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