Calculus Examples

$y = \ln (| 3 x^{2} - 7 x |)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (| 3 x^{2} - 7 x |))$

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 $f (x) = \ln (x)$ and $g (x) = | 3 x^{2} - 7 x |$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $| 3 x^{2} - 7 x |$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| 3 x^{2} - 7 x |]$

Step 3.1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [| 3 x^{2} - 7 x |]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $| 3 x^{2} - 7 x |$ .

$\frac{1}{| 3 x^{2} - 7 x |} \frac{d}{d x} [| 3 x^{2} - 7 x |]$

Step 3.2

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 $g (x) = 3 x^{2} - 7 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $3 x^{2} - 7 x$ .

$\frac{1}{| 3 x^{2} - 7 x |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [3 x^{2} - 7 x])$

Step 3.2.2

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

$\frac{1}{| 3 x^{2} - 7 x |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [3 x^{2} - 7 x])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $3 x^{2} - 7 x$ .

$\frac{1}{| 3 x^{2} - 7 x |} (\frac{3 x^{2} - 7 x}{| 3 x^{2} - 7 x |} \frac{d}{d x} [3 x^{2} - 7 x])$

Step 3.3

Multiply $\frac{3 x^{2} - 7 x}{| 3 x^{2} - 7 x |}$ by $\frac{1}{| 3 x^{2} - 7 x |}$ .

$\frac{3 x^{2} - 7 x}{| 3 x^{2} - 7 x | | 3 x^{2} - 7 x |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.4

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

$\frac{3 x^{2} - 7 x}{| (3 x^{2} - 7 x) (3 x^{2} - 7 x) |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.5

Raise $3 x^{2} - 7 x$ to the power of $1$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{1} (3 x^{2} - 7 x) |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.6

Raise $3 x^{2} - 7 x$ to the power of $1$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{1} {(3 x^{2} - 7 x)}^{1} |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.7

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

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{1 + 1} |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.8

Add $1$ and $1$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} \frac{d}{d x} [3 x^{2} - 7 x]$

Step 3.9

By the Sum Rule, the derivative of $3 x^{2} - 7 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x]$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x])$

Step 3.10

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x])$

Step 3.11

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

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (3 (2 x) + \frac{d}{d x} [- 7 x])$

Step 3.12

Multiply $2$ by $3$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (6 x + \frac{d}{d x} [- 7 x])$

Step 3.13

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (6 x - 7 \frac{d}{d x} [x])$

Step 3.14

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

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (6 x - 7 \cdot 1)$

Step 3.15

Multiply $- 7$ by $1$ .

$\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (6 x - 7)$

Step 3.16

Simplify.

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Step 3.16.1

Reorder the factors of $\frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |} (6 x - 7)$ .

$(6 x - 7) \frac{3 x^{2} - 7 x}{| {(3 x^{2} - 7 x)}^{2} |}$

Step 3.16.2

Factor $x$ out of $3 x^{2} - 7 x$ .

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Step 3.16.2.1

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

$(6 x - 7) \frac{x (3 x) - 7 x}{| {(3 x^{2} - 7 x)}^{2} |}$

Step 3.16.2.2

Factor $x$ out of $- 7 x$ .

$(6 x - 7) \frac{x (3 x) + x \cdot - 7}{| {(3 x^{2} - 7 x)}^{2} |}$

Step 3.16.2.3

Factor $x$ out of $x (3 x) + x \cdot - 7$ .

$(6 x - 7) \frac{x (3 x - 7)}{| {(3 x^{2} - 7 x)}^{2} |}$

Step 3.16.3

Simplify the denominator.

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Step 3.16.3.1

Factor $x$ out of $3 x^{2} - 7 x$ .

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Step 3.16.3.1.1

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

$(6 x - 7) \frac{x (3 x - 7)}{| {(x (3 x) - 7 x)}^{2} |}$

Step 3.16.3.1.2

Factor $x$ out of $- 7 x$ .

$(6 x - 7) \frac{x (3 x - 7)}{| {(x (3 x) + x \cdot - 7)}^{2} |}$

Step 3.16.3.1.3

Factor $x$ out of $x (3 x) + x \cdot - 7$ .

$(6 x - 7) \frac{x (3 x - 7)}{| {(x (3 x - 7))}^{2} |}$

Step 3.16.3.2

Apply the product rule to $x (3 x - 7)$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} {(3 x - 7)}^{2} |}$

Step 3.16.3.3

Rewrite ${(3 x - 7)}^{2}$ as $(3 x - 7) (3 x - 7)$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} ((3 x - 7) (3 x - 7)) |}$

Step 3.16.3.4

Expand $(3 x - 7) (3 x - 7)$ using the FOIL Method.

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Step 3.16.3.4.1

Apply the distributive property.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 x (3 x - 7) - 7 (3 x - 7)) |}$

Step 3.16.3.4.2

Apply the distributive property.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 x (3 x) + 3 x \cdot - 7 - 7 (3 x - 7)) |}$

Step 3.16.3.4.3

Apply the distributive property.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 x (3 x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5

Simplify and combine like terms.

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Step 3.16.3.5.1

Simplify each term.

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Step 3.16.3.5.1.1

Rewrite using the commutative property of multiplication.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 \cdot 3 x \cdot x + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5.1.2

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

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Step 3.16.3.5.1.2.1

Move $x$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 \cdot 3 (x \cdot x) + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5.1.2.2

Multiply $x$ by $x$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (3 \cdot 3 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5.1.3

Multiply $3$ by $3$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} + 3 x \cdot - 7 - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5.1.4

Multiply $- 7$ by $3$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 21 x - 7 (3 x) - 7 \cdot - 7) |}$

Step 3.16.3.5.1.5

Multiply $3$ by $- 7$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 21 x - 21 x - 7 \cdot - 7) |}$

Step 3.16.3.5.1.6

Multiply $- 7$ by $- 7$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 21 x - 21 x + 49) |}$

Step 3.16.3.5.2

Subtract $21 x$ from $- 21 x$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 42 x + 49) |}$

Step 3.16.3.6

Apply the distributive property.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2}) + x^{2} (- 42 x) + x^{2} \cdot 49 |}$

Step 3.16.3.7

Simplify.

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Step 3.16.3.7.1

Rewrite using the commutative property of multiplication.

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{2} x^{2} + x^{2} (- 42 x) + x^{2} \cdot 49 |}$

Step 3.16.3.7.2

Rewrite using the commutative property of multiplication.

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{2} x^{2} - 42 x^{2} x + x^{2} \cdot 49 |}$

Step 3.16.3.7.3

Move $49$ to the left of $x^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{2} x^{2} - 42 x^{2} x + 49 \cdot x^{2} |}$

Step 3.16.3.8

Simplify each term.

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Step 3.16.3.8.1

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

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Step 3.16.3.8.1.1

Move $x^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| 9 (x^{2} x^{2}) - 42 x^{2} x + 49 \cdot x^{2} |}$

Step 3.16.3.8.1.2

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

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{2 + 2} - 42 x^{2} x + 49 \cdot x^{2} |}$

Step 3.16.3.8.1.3

Add $2$ and $2$ .

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 x^{2} x + 49 \cdot x^{2} |}$

Step 3.16.3.8.2

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

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Step 3.16.3.8.2.1

Move $x$ .

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 (x \cdot x^{2}) + 49 \cdot x^{2} |}$

Step 3.16.3.8.2.2

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

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Step 3.16.3.8.2.2.1

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

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 (x^{1} x^{2}) + 49 \cdot x^{2} |}$

Step 3.16.3.8.2.2.2

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

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 x^{1 + 2} + 49 \cdot x^{2} |}$

Step 3.16.3.8.2.3

Add $1$ and $2$ .

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 x^{3} + 49 \cdot x^{2} |}$

$(6 x - 7) \frac{x (3 x - 7)}{| 9 x^{4} - 42 x^{3} + 49 x^{2} |}$

Step 3.16.3.9

Factor $x^{2}$ out of $9 x^{4} - 42 x^{3} + 49 x^{2}$ .

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Step 3.16.3.9.1

Factor $x^{2}$ out of $9 x^{4}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2}) - 42 x^{3} + 49 x^{2} |}$

Step 3.16.3.9.2

Factor $x^{2}$ out of $- 42 x^{3}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2}) + x^{2} (- 42 x) + 49 x^{2} |}$

Step 3.16.3.9.3

Factor $x^{2}$ out of $49 x^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2}) + x^{2} (- 42 x) + x^{2} \cdot 49 |}$

Step 3.16.3.9.4

Factor $x^{2}$ out of $x^{2} (9 x^{2}) + x^{2} (- 42 x)$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 42 x) + x^{2} \cdot 49 |}$

Step 3.16.3.9.5

Factor $x^{2}$ out of $x^{2} (9 x^{2} - 42 x) + x^{2} \cdot 49$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} (9 x^{2} - 42 x + 49) |}$

Step 3.16.3.10

Factor using the perfect square rule.

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Step 3.16.3.10.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} ({(3 x)}^{2} - 42 x + 49) |}$

Step 3.16.3.10.2

Rewrite $49$ as $7^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} ({(3 x)}^{2} - 42 x + 7^{2}) |}$

Step 3.16.3.10.3

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

$42 x = 2 \cdot (3 x) \cdot 7$

Step 3.16.3.10.4

Rewrite the polynomial.

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} ({(3 x)}^{2} - 2 \cdot (3 x) \cdot 7 + 7^{2}) |}$

Step 3.16.3.10.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 x$ and $b = 7$ .

$(6 x - 7) \frac{x (3 x - 7)}{| x^{2} {(3 x - 7)}^{2} |}$

Step 3.16.4

Remove non-negative terms from the absolute value.

$(6 x - 7) \frac{x (3 x - 7)}{x^{2} {(3 x - 7)}^{2}}$

Step 3.16.5

Cancel the common factors.

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Step 3.16.5.1

Factor $x$ out of $x^{2} {(3 x - 7)}^{2}$ .

$(6 x - 7) \frac{x (3 x - 7)}{x (x {(3 x - 7)}^{2})}$

Step 3.16.5.2

Cancel the common factor.

$(6 x - 7) \frac{x (3 x - 7)}{x (x {(3 x - 7)}^{2})}$

Step 3.16.5.3

Rewrite the expression.

$(6 x - 7) \frac{3 x - 7}{x {(3 x - 7)}^{2}}$

Step 3.16.6

Cancel the common factor of $3 x - 7$ and ${(3 x - 7)}^{2}$ .

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Step 3.16.6.1

Multiply by $1$ .

$(6 x - 7) \frac{(3 x - 7) \cdot 1}{x {(3 x - 7)}^{2}}$

Step 3.16.6.2

Cancel the common factors.

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Step 3.16.6.2.1

Factor $3 x - 7$ out of $x {(3 x - 7)}^{2}$ .

$(6 x - 7) \frac{(3 x - 7) \cdot 1}{(3 x - 7) (x (3 x - 7))}$

Step 3.16.6.2.2

Cancel the common factor.

$(6 x - 7) \frac{(3 x - 7) \cdot 1}{(3 x - 7) (x (3 x - 7))}$

Step 3.16.6.2.3

Rewrite the expression.

$(6 x - 7) \frac{1}{x (3 x - 7)}$

Step 3.16.7

Multiply $6 x - 7$ by $\frac{1}{x (3 x - 7)}$ .

$\frac{6 x - 7}{x (3 x - 7)}$

Step 4

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

$y' = \frac{6 x - 7}{x (3 x - 7)}$

Step 5

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

$\frac{d y}{d x} = \frac{6 x - 7}{x (3 x - 7)}$