Calculus Examples

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

Step 1

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

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $2$ .

$2 \frac{x^{2} \frac{d}{d x} [\ln (x + 3)] - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

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 $g (x) = x + 3$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$2 \frac{x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 3]) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

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

$2 \frac{x^{2} (\frac{1}{u} \frac{d}{d x} [x + 3]) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x + 3$ .

$2 \frac{x^{2} (\frac{1}{x + 3} \frac{d}{d x} [x + 3]) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5

Differentiate.

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Step 5.1

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

$2 \frac{\frac{x^{2}}{x + 3} \frac{d}{d x} [x + 3] - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.2

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

$2 \frac{\frac{x^{2}}{x + 3} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.3

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

$2 \frac{\frac{x^{2}}{x + 3} (1 + \frac{d}{d x} [3]) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 \frac{\frac{x^{2}}{x + 3} (1 + 0) - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $1$ and $0$ .

$2 \frac{\frac{x^{2}}{x + 3} \cdot 1 - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 5.5.2

Multiply $\frac{x^{2}}{x + 3}$ by $1$ .

$2 \frac{\frac{x^{2}}{x + 3} - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6

Multiply $\frac{\frac{x^{2}}{x + 3} - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}}$ by $\frac{x + 3}{x + 3}$ .

$2 (\frac{x + 3}{x + 3} \cdot \frac{\frac{x^{2}}{x + 3} - \ln (x + 3) \frac{d}{d x} [x^{2}]}{x^{4}})$

Step 7

Simplify terms.

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Step 7.1

Combine.

$2 \frac{(x + 3) (\frac{x^{2}}{x + 3} - \ln (x + 3) \frac{d}{d x} [x^{2}])}{(x + 3) x^{4}}$

Step 7.2

Apply the distributive property.

$2 \frac{(x + 3) \frac{x^{2}}{x + 3} + (x + 3) (- \ln (x + 3) \frac{d}{d x} [x^{2}])}{(x + 3) x^{4}}$

Step 7.3

Cancel the common factor of $x + 3$ .

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Step 7.3.1

Cancel the common factor.

$2 \frac{(x + 3) \frac{x^{2}}{x + 3} + (x + 3) (- \ln (x + 3) \frac{d}{d x} [x^{2}])}{(x + 3) x^{4}}$

Step 7.3.2

Rewrite the expression.

$2 \frac{x^{2} + (x + 3) (- \ln (x + 3) \frac{d}{d x} [x^{2}])}{(x + 3) x^{4}}$

Step 8

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

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

Step 9

Combine fractions.

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Step 9.1

Multiply $2$ by $- 1$ .

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

Step 9.2

Combine $2$ and $\frac{x^{2} + (x + 3) (- 2 \ln (x + 3) x)}{(x + 3) x^{4}}$ .

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Apply the distributive property.

$\frac{2 x^{2} + 2 ((x + 3) (- 2 \ln (x + 3) x))}{x \cdot x^{4} + 3 x^{4}}$

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 $- 2 \ln (x + 3)$ by moving $2$ inside the logarithm.

$\frac{2 x^{2} + 2 ((x + 3) (- \ln ({(x + 3)}^{2}) x))}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.2

Apply the distributive property.

$\frac{2 x^{2} + 2 (x (- \ln ({(x + 3)}^{2}) x) + 3 (- \ln ({(x + 3)}^{2}) x))}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} + 2 (- x (\ln ({(x + 3)}^{2}) x) + 3 (- \ln ({(x + 3)}^{2}) x))}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.4

Multiply $3 (- \ln ({(x + 3)}^{2}) x)$ .

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Step 10.3.1.4.1

Multiply $- 1$ by $3$ .

$\frac{2 x^{2} + 2 (- x (\ln ({(x + 3)}^{2}) x) - 3 (\ln ({(x + 3)}^{2}) x))}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.4.2

Reorder $\ln ({(x + 3)}^{2})$ and $- 3$ .

$\frac{2 x^{2} + 2 (- x (\ln ({(x + 3)}^{2}) x) - 3 \cdot \ln ({(x + 3)}^{2}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.4.3

Simplify $- 3 \cdot \ln ({(x + 3)}^{2})$ by moving $3$ inside the logarithm.

$\frac{2 x^{2} + 2 (- x (\ln ({(x + 3)}^{2}) x) - \ln ({({(x + 3)}^{2})}^{3}) x)}{x \cdot x^{4} + 3 x^{4}}$

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 $x$ .

$\frac{2 x^{2} + 2 (- (x \cdot x) \ln ({(x + 3)}^{2}) - \ln ({({(x + 3)}^{2})}^{3}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.5.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 (- x^{2} \ln ({(x + 3)}^{2}) - \ln ({({(x + 3)}^{2})}^{3}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.5.2

Multiply the exponents in ${({(x + 3)}^{2})}^{3}$ .

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Step 10.3.1.5.2.1

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

$\frac{2 x^{2} + 2 (- x^{2} \ln ({(x + 3)}^{2}) - \ln ({(x + 3)}^{2 \cdot 3}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.5.2.2

Multiply $2$ by $3$ .

$\frac{2 x^{2} + 2 (- x^{2} \ln ({(x + 3)}^{2}) - \ln ({(x + 3)}^{6}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.6

Apply the distributive property.

$\frac{2 x^{2} + 2 (- x^{2} \ln ({(x + 3)}^{2})) + 2 (- \ln ({(x + 3)}^{6}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.7

Multiply $2 (- x^{2} \ln ({(x + 3)}^{2}))$ .

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Step 10.3.1.7.1

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} - 2 (x^{2} \ln ({(x + 3)}^{2})) + 2 (- \ln ({(x + 3)}^{6}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.7.2

Simplify $- 2 \ln ({(x + 3)}^{2})$ by moving $2$ inside the logarithm.

$\frac{2 x^{2} + x^{2} (- \ln ({({(x + 3)}^{2})}^{2})) + 2 (- \ln ({(x + 3)}^{6}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.8

Multiply $2 (- \ln ({(x + 3)}^{6}) x)$ .

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Step 10.3.1.8.1

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} + x^{2} (- \ln ({({(x + 3)}^{2})}^{2})) - 2 (\ln ({(x + 3)}^{6}) x)}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.8.2

Reorder $\ln ({(x + 3)}^{6})$ and $- 2$ .

$\frac{2 x^{2} + x^{2} (- \ln ({({(x + 3)}^{2})}^{2})) - 2 \cdot \ln ({(x + 3)}^{6}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.8.3

Simplify $- 2 \cdot \ln ({(x + 3)}^{6})$ by moving $2$ inside the logarithm.

$\frac{2 x^{2} + x^{2} (- \ln ({({(x + 3)}^{2})}^{2})) - \ln ({({(x + 3)}^{6})}^{2}) x}{x \cdot x^{4} + 3 x^{4}}$

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.

$\frac{2 x^{2} - x^{2} \ln ({({(x + 3)}^{2})}^{2}) - \ln ({({(x + 3)}^{6})}^{2}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.9.2

Multiply the exponents in ${({(x + 3)}^{2})}^{2}$ .

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Step 10.3.1.9.2.1

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

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{2 \cdot 2}) - \ln ({({(x + 3)}^{6})}^{2}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.9.2.2

Multiply $2$ by $2$ .

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - \ln ({({(x + 3)}^{6})}^{2}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.9.3

Multiply the exponents in ${({(x + 3)}^{6})}^{2}$ .

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Step 10.3.1.9.3.1

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

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{6 \cdot 2}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.1.9.3.2

Multiply $6$ by $2$ .

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}) x}{x \cdot x^{4} + 3 x^{4}}$

Step 10.3.2

Reorder factors in $2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}) x$ .

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})}{x \cdot x^{4} + 3 x^{4}}$

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 $1$ .

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})}{x^{1} x^{4} + 3 x^{4}}$

Step 10.4.1.2

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

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})}{x^{1 + 4} + 3 x^{4}}$

Step 10.4.2

Add $1$ and $4$ .

$\frac{2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})}{x^{5} + 3 x^{4}}$

Step 10.5

Factor $x$ out of $2 x^{2} - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})$ .

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Step 10.5.1

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

$\frac{x (2 x) - x^{2} \ln ({(x + 3)}^{4}) - x \ln ({(x + 3)}^{12})}{x^{5} + 3 x^{4}}$

Step 10.5.2

Factor $x$ out of $- x^{2} \ln ({(x + 3)}^{4})$ .

$\frac{x (2 x) + x (- x \ln ({(x + 3)}^{4})) - x \ln ({(x + 3)}^{12})}{x^{5} + 3 x^{4}}$

Step 10.5.3

Factor $x$ out of $- x \ln ({(x + 3)}^{12})$ .

$\frac{x (2 x) + x (- x \ln ({(x + 3)}^{4})) + x (- \ln ({(x + 3)}^{12}))}{x^{5} + 3 x^{4}}$

Step 10.5.4

Factor $x$ out of $x (2 x) + x (- x \ln ({(x + 3)}^{4}))$ .

$\frac{x (2 x - x \ln ({(x + 3)}^{4})) + x (- \ln ({(x + 3)}^{12}))}{x^{5} + 3 x^{4}}$

Step 10.5.5

Factor $x$ out of $x (2 x - x \ln ({(x + 3)}^{4})) + x (- \ln ({(x + 3)}^{12}))$ .

$\frac{x (2 x - x \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}))}{x^{5} + 3 x^{4}}$

Step 10.6

Factor $x^{4}$ out of $x^{5} + 3 x^{4}$ .

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Step 10.6.1

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

$\frac{x (2 x - x \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}))}{x^{4} x + 3 x^{4}}$

Step 10.6.2

Factor $x^{4}$ out of $3 x^{4}$ .

$\frac{x (2 x - x \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}))}{x^{4} x + x^{4} \cdot 3}$

Step 10.6.3

Factor $x^{4}$ out of $x^{4} x + x^{4} \cdot 3$ .

$\frac{x (2 x - x \ln ({(x + 3)}^{4}) - \ln ({(x + 3)}^{12}))}{x^{4} (x + 3)}$

Step 10.7

Expand $\ln ({(x + 3)}^{4})$ by moving $4$ outside the logarithm.

$\frac{x (2 x - x (4 \ln (x + 3)) - \ln ({(x + 3)}^{12}))}{x^{4} (x + 3)}$

Step 10.8

Expand $\ln ({(x + 3)}^{12})$ by moving $12$ outside the logarithm.

$\frac{x (2 x - x (4 \ln (x + 3)) - (12 \ln (x + 3)))}{x^{4} (x + 3)}$

Step 10.9

Cancel the common factors.

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Step 10.9.1

Factor $x$ out of $x^{4} (x + 3)$ .

$\frac{x (2 x - x (4 \ln (x + 3)) - (12 \ln (x + 3)))}{x (x^{3} (x + 3))}$

Step 10.9.2

Cancel the common factor.

$\frac{x (2 x - x (4 \ln (x + 3)) - (12 \ln (x + 3)))}{x (x^{3} (x + 3))}$

Step 10.9.3

Rewrite the expression.

$\frac{2 x - x (4 \ln (x + 3)) - (12 \ln (x + 3))}{x^{3} (x + 3)}$

Step 10.10

Simplify the numerator.

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Step 10.10.1

Multiply $4$ by $- 1$ .

$\frac{2 x - 4 x \ln (x + 3) - 1 \cdot 12 \ln (x + 3)}{x^{3} (x + 3)}$

Step 10.10.2

Multiply $- 1$ by $12$ .

$\frac{2 x - 4 x \ln (x + 3) - 12 \ln (x + 3)}{x^{3} (x + 3)}$

Step 10.10.3

Factor $2$ out of $2 x - 4 x \ln (x + 3) - 12 \ln (x + 3)$ .

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Step 10.10.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4 x \ln (x + 3) - 12 \ln (x + 3)}{x^{3} (x + 3)}$

Step 10.10.3.2

Factor $2$ out of $- 4 x \ln (x + 3)$ .

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

Step 10.10.3.3

Factor $2$ out of $- 12 \ln (x + 3)$ .

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

Step 10.10.3.4

Factor $2$ out of $2 (x) + 2 (- 2 x \ln (x + 3))$ .

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

Step 10.10.3.5

Factor $2$ out of $2 (x - 2 x \ln (x + 3)) + 2 (- 6 \ln (x + 3))$ .

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