Calculus Examples

$y = \frac{5 x^{2}}{\ln (| 3 x |)}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x^{2}}{\ln (| 3 x |)}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x^{2}}{\ln (| 3 x |)}]$ .

$5 \frac{d}{d x} [\frac{x^{2}}{\ln (| 3 x |)}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = \ln (| 3 x |)$ .

$5 \frac{\ln (| 3 x |) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [\ln (| 3 x |)]}{\ln^{2} (| 3 x |)}$

Step 3

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

$5 \frac{\ln (| 3 x |) (2 x) - x^{2} \frac{d}{d x} [\ln (| 3 x |)]}{\ln^{2} (| 3 x |)}$

Step 4

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 $g (x) = | 3 x |$ .

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Step 4.1

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

$5 \frac{\ln (| 3 x |) (2 x) - x^{2} (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| 3 x |])}{\ln^{2} (| 3 x |)}$

Step 4.2

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

$5 \frac{\ln (| 3 x |) (2 x) - x^{2} (\frac{1}{u_{1}} \frac{d}{d x} [| 3 x |])}{\ln^{2} (| 3 x |)}$

Step 4.3

Replace all occurrences of $u_{1}$ with $| 3 x |$ .

$5 \frac{\ln (| 3 x |) (2 x) - x^{2} (\frac{1}{| 3 x |} \frac{d}{d x} [| 3 x |])}{\ln^{2} (| 3 x |)}$

Step 5

Combine $\frac{1}{| 3 x |}$ and $x^{2}$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{x^{2}}{| 3 x |} \frac{d}{d x} [| 3 x |]}{\ln^{2} (| 3 x |)}$

Step 6

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 $g (x) = 3 x$ .

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Step 6.1

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{x^{2}}{| 3 x |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [3 x])}{\ln^{2} (| 3 x |)}$

Step 6.2

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{x^{2}}{| 3 x |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [3 x])}{\ln^{2} (| 3 x |)}$

Step 6.3

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{x^{2}}{| 3 x |} (\frac{3 x}{| 3 x |} \frac{d}{d x} [3 x])}{\ln^{2} (| 3 x |)}$

Step 7

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x \cdot x^{2}}{| 3 x | | 3 x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 8

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

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Step 8.1

Move $x^{2}$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 (x^{2} x)}{| 3 x | | 3 x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 8.2

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

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Step 8.2.1

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 (x^{2} x^{1})}{| 3 x | | 3 x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 8.2.2

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{2 + 1}}{| 3 x | | 3 x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 8.3

Add $2$ and $1$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 3 x | | 3 x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 9

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 3 x (3 x) |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 10

Multiply $3$ by $3$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 x \cdot x |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 11

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 (x^{1} x) |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 12

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 (x^{1} x^{1}) |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 13

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 x^{1 + 1} |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 14

Add $1$ and $1$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 x^{2} |} \frac{d}{d x} [3 x]}{\ln^{2} (| 3 x |)}$

Step 15

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{3 x^{3}}{| 9 x^{2} |} (3 \frac{d}{d x} [x])}{\ln^{2} (| 3 x |)}$

Step 16

Combine fractions.

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Step 16.1

Multiply $3$ by $- 1$ .

$5 \frac{\ln (| 3 x |) (2 x) - 3 \frac{3 x^{3}}{| 9 x^{2} |} \frac{d}{d x} [x]}{\ln^{2} (| 3 x |)}$

Step 16.2

Combine $- 3$ and $\frac{3 x^{3}}{| 9 x^{2} |}$ .

$5 \frac{\ln (| 3 x |) (2 x) + \frac{- 3 (3 x^{3})}{| 9 x^{2} |} \frac{d}{d x} [x]}{\ln^{2} (| 3 x |)}$

Step 16.3

Simplify the expression.

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Step 16.3.1

Multiply $3$ by $- 3$ .

$5 \frac{\ln (| 3 x |) (2 x) + \frac{- 9 x^{3}}{| 9 x^{2} |} \frac{d}{d x} [x]}{\ln^{2} (| 3 x |)}$

Step 16.3.2

Move the negative in front of the fraction.

$5 \frac{\ln (| 3 x |) (2 x) - \frac{9 x^{3}}{| 9 x^{2} |} \frac{d}{d x} [x]}{\ln^{2} (| 3 x |)}$

Step 17

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

$5 \frac{\ln (| 3 x |) (2 x) - \frac{9 x^{3}}{| 9 x^{2} |} \cdot 1}{\ln^{2} (| 3 x |)}$

Step 18

Combine fractions.

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Step 18.1

Multiply $- 1$ by $1$ .

$5 \frac{\ln (| 3 x |) (2 x) - \frac{9 x^{3}}{| 9 x^{2} |}}{\ln^{2} (| 3 x |)}$

Step 18.2

Combine $5$ and $\frac{\ln (| 3 x |) (2 x) - \frac{9 x^{3}}{| 9 x^{2} |}}{\ln^{2} (| 3 x |)}$ .

$\frac{5 (\ln (| 3 x |) (2 x) - \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$\frac{5 (\ln (| 3 x |) (2 x)) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2

Simplify the numerator.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 (2 \ln (| 3 x |) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.2

Remove non-negative terms from the absolute value.

$\frac{5 (2 \ln (3 | x |) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.3

Simplify $2 \ln (3 | x |)$ by moving $2$ inside the logarithm.

$\frac{5 (\ln ({(3 | x |)}^{2}) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.4

Apply the product rule to $3 | x |$ .

$\frac{5 (\ln (3^{2} {| x |}^{2}) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.5

Raise $3$ to the power of $2$ .

$\frac{5 (\ln (9 {| x |}^{2}) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.6

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$\frac{5 (\ln (9 x^{2}) x) + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.7

Multiply $5 (\ln (9 x^{2}) x)$ .

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Step 19.2.1.7.1

Reorder $\ln (9 x^{2})$ and $5$ .

$\frac{5 \cdot \ln (9 x^{2}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.7.2

Simplify $5 \cdot \ln (9 x^{2})$ by moving $5$ inside the logarithm.

$\frac{\ln ({(9 x^{2})}^{5}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.8

Apply the product rule to $9 x^{2}$ .

$\frac{\ln (9^{5} {(x^{2})}^{5}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.9

Raise $9$ to the power of $5$ .

$\frac{\ln (59049 {(x^{2})}^{5}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.10

Multiply the exponents in ${(x^{2})}^{5}$ .

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Step 19.2.1.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\ln (59049 x^{2 \cdot 5}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.10.2

Multiply $2$ by $5$ .

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{9 x^{3}}{| 9 x^{2} |})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.11

Remove non-negative terms from the absolute value.

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{9 x^{3}}{9 x^{2}})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.12

Cancel the common factor of $9$ .

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Step 19.2.1.12.1

Cancel the common factor.

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{9 x^{3}}{9 x^{2}})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.12.2

Rewrite the expression.

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{x^{3}}{x^{2}})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.13

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 19.2.1.13.1

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

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{x^{2} x}{x^{2}})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.13.2

Cancel the common factors.

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Step 19.2.1.13.2.1

Multiply by $1$ .

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{x^{2} x}{x^{2} \cdot 1})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.13.2.2

Cancel the common factor.

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{x^{2} x}{x^{2} \cdot 1})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.13.2.3

Rewrite the expression.

$\frac{\ln (59049 x^{10}) x + 5 (- \frac{x}{1})}{\ln^{2} (| 3 x |)}$

Step 19.2.1.13.2.4

Divide $x$ by $1$ .

$\frac{\ln (59049 x^{10}) x + 5 (- x)}{\ln^{2} (| 3 x |)}$

Step 19.2.1.14

Multiply $- 1$ by $5$ .

$\frac{\ln (59049 x^{10}) x - 5 x}{\ln^{2} (| 3 x |)}$

Step 19.2.2

Reorder factors in $\ln (59049 x^{10}) x - 5 x$ .

$\frac{x \ln (59049 x^{10}) - 5 x}{\ln^{2} (| 3 x |)}$

Step 19.3

Factor $x$ out of $x \ln (59049 x^{10}) - 5 x$ .

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Step 19.3.1

Factor $x$ out of $x \ln (59049 x^{10})$ .

$\frac{x (\ln (59049 x^{10})) - 5 x}{\ln^{2} (| 3 x |)}$

Step 19.3.2

Factor $x$ out of $- 5 x$ .

$\frac{x (\ln (59049 x^{10})) + x \cdot - 5}{\ln^{2} (| 3 x |)}$

Step 19.3.3

Factor $x$ out of $x (\ln (59049 x^{10})) + x \cdot - 5$ .

$\frac{x (\ln (59049 x^{10}) - 5)}{\ln^{2} (| 3 x |)}$

Step 19.4

Remove non-negative terms from the absolute value.

$\frac{x (\ln (59049 x^{10}) - 5)}{\ln^{2} (3 | x |)}$