Calculus Examples

$\frac{9 \ln (x + 6)}{x^{2}}$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9 \ln (x + 6)}{x^{2}}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{\ln (x + 6)}{x^{2}}]$ .

$9 \frac{d}{d x} [\frac{\ln (x + 6)}{x^{2}}]$

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 $f (x) = \ln (x + 6)$ and $g (x) = x^{2}$ .

$9 \frac{x^{2} \frac{d}{d x} [\ln (x + 6)] - \ln (x + 6) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3

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

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Step 3.1

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

$9 \frac{x^{2} \frac{d}{d x} [\ln (x + 6)] - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 3.2

Multiply $2$ by $2$ .

$9 \frac{x^{2} \frac{d}{d x} [\ln (x + 6)] - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

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 $g (x) = x + 6$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x + 6$ .

$9 \frac{x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 6]) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

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

$9 \frac{x^{2} (\frac{1}{u} \frac{d}{d x} [x + 6]) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x + 6$ .

$9 \frac{x^{2} (\frac{1}{x + 6} \frac{d}{d x} [x + 6]) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{x + 6}$ and $x^{2}$ .

$9 \frac{\frac{x^{2}}{x + 6} \frac{d}{d x} [x + 6] - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$9 \frac{\frac{x^{2}}{x + 6} (\frac{d}{d x} [x] + \frac{d}{d x} [6]) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.3

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

$9 \frac{\frac{x^{2}}{x + 6} (1 + \frac{d}{d x} [6]) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$9 \frac{\frac{x^{2}}{x + 6} (1 + 0) - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $1$ and $0$ .

$9 \frac{\frac{x^{2}}{x + 6} \cdot 1 - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.5.2

Multiply $\frac{x^{2}}{x + 6}$ by $1$ .

$9 \frac{\frac{x^{2}}{x + 6} - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6

Multiply $\frac{\frac{x^{2}}{x + 6} - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}}$ by $\frac{x + 6}{x + 6}$ .

$9 (\frac{x + 6}{x + 6} \cdot \frac{\frac{x^{2}}{x + 6} - \ln (x + 6) \frac{d}{d x} [x^{2}]}{x^{4}})$

Step 7

Simplify terms.

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Step 7.1

Combine.

$9 \frac{(x + 6) (\frac{x^{2}}{x + 6} - \ln (x + 6) \frac{d}{d x} [x^{2}])}{(x + 6) x^{4}}$

Step 7.2

Apply the distributive property.

$9 \frac{(x + 6) \frac{x^{2}}{x + 6} + (x + 6) (- \ln (x + 6) \frac{d}{d x} [x^{2}])}{(x + 6) x^{4}}$

Step 7.3

Cancel the common factor of $x + 6$ .

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Step 7.3.1

Cancel the common factor.

$9 \frac{(x + 6) \frac{x^{2}}{x + 6} + (x + 6) (- \ln (x + 6) \frac{d}{d x} [x^{2}])}{(x + 6) x^{4}}$

Step 7.3.2

Rewrite the expression.

$9 \frac{x^{2} + (x + 6) (- \ln (x + 6) \frac{d}{d x} [x^{2}])}{(x + 6) x^{4}}$

Step 8

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

$9 \frac{x^{2} + (x + 6) (- \ln (x + 6) (2 x))}{(x + 6) x^{4}}$

Step 9

Combine fractions.

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Step 9.1

Multiply $2$ by $- 1$ .

$9 \frac{x^{2} + (x + 6) (- 2 \ln (x + 6) x)}{(x + 6) x^{4}}$

Step 9.2

Combine $9$ and $\frac{x^{2} + (x + 6) (- 2 \ln (x + 6) x)}{(x + 6) x^{4}}$ .

$\frac{9 (x^{2} + (x + 6) (- 2 \ln (x + 6) x))}{(x + 6) x^{4}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{9 x^{2} + 9 ((x + 6) (- 2 \ln (x + 6) x))}{(x + 6) x^{4}}$

Step 10.2

Apply the distributive property.

$\frac{9 x^{2} + 9 ((x + 6) (- 2 \ln (x + 6) x))}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Simplify $- 2 \ln (x + 6)$ by moving $2$ inside the logarithm.

$\frac{9 x^{2} + 9 ((x + 6) (- \ln ({(x + 6)}^{2}) x))}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.2

Apply the distributive property.

$\frac{9 x^{2} + 9 (x (- \ln ({(x + 6)}^{2}) x) + 6 (- \ln ({(x + 6)}^{2}) x))}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{9 x^{2} + 9 (- x (\ln ({(x + 6)}^{2}) x) + 6 (- \ln ({(x + 6)}^{2}) x))}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.4

Multiply $6 (- \ln ({(x + 6)}^{2}) x)$ .

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Step 10.3.1.4.1

Multiply $- 1$ by $6$ .

$\frac{9 x^{2} + 9 (- x (\ln ({(x + 6)}^{2}) x) - 6 (\ln ({(x + 6)}^{2}) x))}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.4.2

Reorder $\ln ({(x + 6)}^{2})$ and $- 6$ .

$\frac{9 x^{2} + 9 (- x (\ln ({(x + 6)}^{2}) x) - 6 \cdot \ln ({(x + 6)}^{2}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.4.3

Simplify $- 6 \cdot \ln ({(x + 6)}^{2})$ by moving $6$ inside the logarithm.

$\frac{9 x^{2} + 9 (- x (\ln ({(x + 6)}^{2}) x) - \ln ({({(x + 6)}^{2})}^{6}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.5

Simplify each term.

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Step 10.3.1.5.1

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

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Step 10.3.1.5.1.1

Move $x$ .

$\frac{9 x^{2} + 9 (- (x \cdot x) \ln ({(x + 6)}^{2}) - \ln ({({(x + 6)}^{2})}^{6}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.5.1.2

Multiply $x$ by $x$ .

$\frac{9 x^{2} + 9 (- x^{2} \ln ({(x + 6)}^{2}) - \ln ({({(x + 6)}^{2})}^{6}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.5.2

Multiply the exponents in ${({(x + 6)}^{2})}^{6}$ .

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Step 10.3.1.5.2.1

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

$\frac{9 x^{2} + 9 (- x^{2} \ln ({(x + 6)}^{2}) - \ln ({(x + 6)}^{2 \cdot 6}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.5.2.2

Multiply $2$ by $6$ .

$\frac{9 x^{2} + 9 (- x^{2} \ln ({(x + 6)}^{2}) - \ln ({(x + 6)}^{12}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.6

Apply the distributive property.

$\frac{9 x^{2} + 9 (- x^{2} \ln ({(x + 6)}^{2})) + 9 (- \ln ({(x + 6)}^{12}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.7

Multiply $9 (- x^{2} \ln ({(x + 6)}^{2}))$ .

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Step 10.3.1.7.1

Multiply $- 1$ by $9$ .

$\frac{9 x^{2} - 9 (x^{2} \ln ({(x + 6)}^{2})) + 9 (- \ln ({(x + 6)}^{12}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.7.2

Simplify $- 9 \ln ({(x + 6)}^{2})$ by moving $9$ inside the logarithm.

$\frac{9 x^{2} + x^{2} (- \ln ({({(x + 6)}^{2})}^{9})) + 9 (- \ln ({(x + 6)}^{12}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.8

Multiply $9 (- \ln ({(x + 6)}^{12}) x)$ .

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Step 10.3.1.8.1

Multiply $- 1$ by $9$ .

$\frac{9 x^{2} + x^{2} (- \ln ({({(x + 6)}^{2})}^{9})) - 9 (\ln ({(x + 6)}^{12}) x)}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.8.2

Reorder $\ln ({(x + 6)}^{12})$ and $- 9$ .

$\frac{9 x^{2} + x^{2} (- \ln ({({(x + 6)}^{2})}^{9})) - 9 \cdot \ln ({(x + 6)}^{12}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.8.3

Simplify $- 9 \cdot \ln ({(x + 6)}^{12})$ by moving $9$ inside the logarithm.

$\frac{9 x^{2} + x^{2} (- \ln ({({(x + 6)}^{2})}^{9})) - \ln ({({(x + 6)}^{12})}^{9}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.9

Simplify each term.

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Step 10.3.1.9.1

Rewrite using the commutative property of multiplication.

$\frac{9 x^{2} - x^{2} \ln ({({(x + 6)}^{2})}^{9}) - \ln ({({(x + 6)}^{12})}^{9}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.9.2

Multiply the exponents in ${({(x + 6)}^{2})}^{9}$ .

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Step 10.3.1.9.2.1

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

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{2 \cdot 9}) - \ln ({({(x + 6)}^{12})}^{9}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.9.2.2

Multiply $2$ by $9$ .

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - \ln ({({(x + 6)}^{12})}^{9}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.9.3

Multiply the exponents in ${({(x + 6)}^{12})}^{9}$ .

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Step 10.3.1.9.3.1

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

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{12 \cdot 9}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.1.9.3.2

Multiply $12$ by $9$ .

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}) x}{x \cdot x^{4} + 6 x^{4}}$

Step 10.3.2

Reorder factors in $9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}) x$ .

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})}{x \cdot x^{4} + 6 x^{4}}$

Step 10.4

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 10.4.1

Multiply $x$ by $x^{4}$ .

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Step 10.4.1.1

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

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})}{x^{1} x^{4} + 6 x^{4}}$

Step 10.4.1.2

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

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})}{x^{1 + 4} + 6 x^{4}}$

Step 10.4.2

Add $1$ and $4$ .

$\frac{9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})}{x^{5} + 6 x^{4}}$

Step 10.5

Factor $x$ out of $9 x^{2} - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})$ .

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Step 10.5.1

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

$\frac{x (9 x) - x^{2} \ln ({(x + 6)}^{18}) - x \ln ({(x + 6)}^{108})}{x^{5} + 6 x^{4}}$

Step 10.5.2

Factor $x$ out of $- x^{2} \ln ({(x + 6)}^{18})$ .

$\frac{x (9 x) + x (- x \ln ({(x + 6)}^{18})) - x \ln ({(x + 6)}^{108})}{x^{5} + 6 x^{4}}$

Step 10.5.3

Factor $x$ out of $- x \ln ({(x + 6)}^{108})$ .

$\frac{x (9 x) + x (- x \ln ({(x + 6)}^{18})) + x (- \ln ({(x + 6)}^{108}))}{x^{5} + 6 x^{4}}$

Step 10.5.4

Factor $x$ out of $x (9 x) + x (- x \ln ({(x + 6)}^{18}))$ .

$\frac{x (9 x - x \ln ({(x + 6)}^{18})) + x (- \ln ({(x + 6)}^{108}))}{x^{5} + 6 x^{4}}$

Step 10.5.5

Factor $x$ out of $x (9 x - x \ln ({(x + 6)}^{18})) + x (- \ln ({(x + 6)}^{108}))$ .

$\frac{x (9 x - x \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}))}{x^{5} + 6 x^{4}}$

Step 10.6

Factor $x^{4}$ out of $x^{5} + 6 x^{4}$ .

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Step 10.6.1

Factor $x^{4}$ out of $x^{5}$ .

$\frac{x (9 x - x \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}))}{x^{4} x + 6 x^{4}}$

Step 10.6.2

Factor $x^{4}$ out of $6 x^{4}$ .

$\frac{x (9 x - x \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}))}{x^{4} x + x^{4} \cdot 6}$

Step 10.6.3

Factor $x^{4}$ out of $x^{4} x + x^{4} \cdot 6$ .

$\frac{x (9 x - x \ln ({(x + 6)}^{18}) - \ln ({(x + 6)}^{108}))}{x^{4} (x + 6)}$

Step 10.7

Expand $\ln ({(x + 6)}^{18})$ by moving $18$ outside the logarithm.

$\frac{x (9 x - x (18 \ln (x + 6)) - \ln ({(x + 6)}^{108}))}{x^{4} (x + 6)}$

Step 10.8

Expand $\ln ({(x + 6)}^{108})$ by moving $108$ outside the logarithm.

$\frac{x (9 x - x (18 \ln (x + 6)) - (108 \ln (x + 6)))}{x^{4} (x + 6)}$

Step 10.9

Cancel the common factors.

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Step 10.9.1

Factor $x$ out of $x^{4} (x + 6)$ .

$\frac{x (9 x - x (18 \ln (x + 6)) - (108 \ln (x + 6)))}{x (x^{3} (x + 6))}$

Step 10.9.2

Cancel the common factor.

$\frac{x (9 x - x (18 \ln (x + 6)) - (108 \ln (x + 6)))}{x (x^{3} (x + 6))}$

Step 10.9.3

Rewrite the expression.

$\frac{9 x - x (18 \ln (x + 6)) - (108 \ln (x + 6))}{x^{3} (x + 6)}$

Step 10.10

Simplify the numerator.

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Step 10.10.1

Multiply $18$ by $- 1$ .

$\frac{9 x - 18 x \ln (x + 6) - 1 \cdot 108 \ln (x + 6)}{x^{3} (x + 6)}$

Step 10.10.2

Multiply $- 1$ by $108$ .

$\frac{9 x - 18 x \ln (x + 6) - 108 \ln (x + 6)}{x^{3} (x + 6)}$

Step 10.10.3

Factor $9$ out of $9 x - 18 x \ln (x + 6) - 108 \ln (x + 6)$ .

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Step 10.10.3.1

Factor $9$ out of $9 x$ .

$\frac{9 (x) - 18 x \ln (x + 6) - 108 \ln (x + 6)}{x^{3} (x + 6)}$

Step 10.10.3.2

Factor $9$ out of $- 18 x \ln (x + 6)$ .

$\frac{9 (x) + 9 (- 2 x \ln (x + 6)) - 108 \ln (x + 6)}{x^{3} (x + 6)}$

Step 10.10.3.3

Factor $9$ out of $- 108 \ln (x + 6)$ .

$\frac{9 (x) + 9 (- 2 x \ln (x + 6)) + 9 (- 12 \ln (x + 6))}{x^{3} (x + 6)}$

Step 10.10.3.4

Factor $9$ out of $9 (x) + 9 (- 2 x \ln (x + 6))$ .

$\frac{9 (x - 2 x \ln (x + 6)) + 9 (- 12 \ln (x + 6))}{x^{3} (x + 6)}$

Step 10.10.3.5

Factor $9$ out of $9 (x - 2 x \ln (x + 6)) + 9 (- 12 \ln (x + 6))$ .

$\frac{9 (x - 2 x \ln (x + 6) - 12 \ln (x + 6))}{x^{3} (x + 6)}$