Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Multiply $- 1$ by $1$ .

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

Step 3.2.5.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

$2 x - 1 = 0$

$\frac{x^{2} - x}{| x^{2} - x |} = 0$

Step 6.2

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

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Step 6.2.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 6.2.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 6.2.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 6.2.2.2

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

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Step 6.2.2.2.1

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

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

Step 6.2.2.2.2

Simplify the left side.

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Step 6.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

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

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Step 6.3.1

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

$\frac{x^{2} - x}{| x^{2} - x |} = 0$

Step 6.3.2

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

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Step 6.3.2.1

Set the numerator equal to zero.

$x^{2} - x = 0$

Step 6.3.2.2

Solve the equation for $x$ .

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

$x \cdot x - x = 0$

Step 6.3.2.2.1.2

Factor $x$ out of $- x$ .

$x \cdot x + x \cdot - 1 = 0$

Step 6.3.2.2.1.3

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

$x (x - 1) = 0$

Step 6.3.2.2.2

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

$x = 0$

$x - 1 = 0$

Step 6.3.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.2.2.4

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

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Step 6.3.2.2.4.1

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

$x - 1 = 0$

Step 6.3.2.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6.3.2.2.5

The final solution is all the values that make $x (x - 1) = 0$ true.

$x = 0, 1$

Step 6.4

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

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

Step 6.5

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

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

Step 7

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

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

One to any power is one.

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

Step 7.1.3

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Multiply $2$ by $2$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Subtract $2$ from $1$ .

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

Step 7.6

Move the negative in front of the fraction.

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

Step 7.7

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

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

Step 8

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

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

Step 9

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