Calculus Examples

$f (x) = \ln (| x^{2} - x |)$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $| x^{2} - x |$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| x^{2} - x |]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [| x^{2} - x |]$

Step 1.3

Replace all occurrences of $u_{1}$ with $| x^{2} - x |$ .

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

Step 2

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 2.1

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

$\frac{1}{| x^{2} - x |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [x^{2} - x])$

Step 2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{| x^{2} - x |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [x^{2} - x])$

Step 2.3

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

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

Step 3

Multiply $\frac{x^{2} - x}{| x^{2} - x |}$ by $\frac{1}{| x^{2} - x |}$ .

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

Step 4

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 5

Raise $x^{2} - x$ to the power of $1$ .

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

Step 6

Raise $x^{2} - x$ to the power of $1$ .

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

Step 7

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

$\frac{x^{2} - x}{| {(x^{2} - x)}^{1 + 1} |} \frac{d}{d x} [x^{2} - x]$

Step 8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Multiply $- 1$ by $1$ .

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

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

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Step 14.2.1

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

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

Step 14.2.2

Factor $x$ out of $- x$ .

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

Step 14.2.3

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

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

Step 14.3

Simplify the denominator.

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Step 14.3.1

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

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Step 14.3.1.1

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

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

Step 14.3.1.2

Factor $x$ out of $- x$ .

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

Step 14.3.1.3

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

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

Step 14.3.2

Apply the product rule to $x (x - 1)$ .

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

Step 14.3.3

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 14.3.4

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 14.3.4.1

Apply the distributive property.

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

Step 14.3.4.2

Apply the distributive property.

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

Step 14.3.4.3

Apply the distributive property.

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

Step 14.3.5

Simplify and combine like terms.

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Step 14.3.5.1

Simplify each term.

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Step 14.3.5.1.1

Multiply $x$ by $x$ .

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

Step 14.3.5.1.2

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

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

Step 14.3.5.1.3

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

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

Step 14.3.5.1.4

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

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

Step 14.3.5.1.5

Multiply $- 1$ by $- 1$ .

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

Step 14.3.5.2

Subtract $x$ from $- x$ .

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

Step 14.3.6

Apply the distributive property.

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

Step 14.3.7

Simplify.

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Step 14.3.7.1

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

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Step 14.3.7.1.1

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

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

Step 14.3.7.1.2

Add $2$ and $2$ .

$(2 x - 1) \frac{x (x - 1)}{| x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 |}$

Step 14.3.7.2

Rewrite using the commutative property of multiplication.

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 x^{2} x + x^{2} \cdot 1 |}$

Step 14.3.7.3

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

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 x^{2} x + x^{2} |}$

Step 14.3.8

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

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Step 14.3.8.1

Move $x$ .

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 (x \cdot x^{2}) + x^{2} |}$

Step 14.3.8.2

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

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Step 14.3.8.2.1

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

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 (x^{1} x^{2}) + x^{2} |}$

Step 14.3.8.2.2

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

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 x^{1 + 2} + x^{2} |}$

Step 14.3.8.3

Add $1$ and $2$ .

$(2 x - 1) \frac{x (x - 1)}{| x^{4} - 2 x^{3} + x^{2} |}$

Step 14.3.9

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

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Step 14.3.9.1

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

$(2 x - 1) \frac{x (x - 1)}{| x^{2} x^{2} - 2 x^{3} + x^{2} |}$

Step 14.3.9.2

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

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

Step 14.3.9.3

Multiply by $1$ .

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

Step 14.3.9.4

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

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

Step 14.3.9.5

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

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

Step 14.3.10

Factor using the perfect square rule.

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Step 14.3.10.1

Rewrite $1$ as $1^{2}$ .

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

Step 14.3.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 14.3.10.3

Rewrite the polynomial.

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

Step 14.3.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 14.4

Remove non-negative terms from the absolute value.

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

Step 14.5

Cancel the common factors.

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Step 14.5.1

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

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

Step 14.5.2

Cancel the common factor.

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

Step 14.5.3

Rewrite the expression.

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

Step 14.6

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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Step 14.6.1

Multiply by $1$ .

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

Step 14.6.2

Cancel the common factors.

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Step 14.6.2.1

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

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

Step 14.6.2.2

Cancel the common factor.

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

Step 14.6.2.3

Rewrite the expression.

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

Step 14.7

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

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