Calculus Examples

$y = \frac{\ln (2 x)}{x^{6}}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $6$ by $2$ .

$\frac{x^{6} \frac{d}{d x} [\ln (2 x)] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.3

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 $g (x) = 2 x$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{x^{6} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x]) - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.3.2

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

$\frac{x^{6} (\frac{1}{u} \frac{d}{d x} [2 x]) - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.3.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{x^{6} (\frac{1}{2 x} \frac{d}{d x} [2 x]) - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4

Differentiate.

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Step 1.4.1

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

$\frac{\frac{x^{6}}{2 x} \frac{d}{d x} [2 x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.2

Cancel the common factor of $x^{6}$ and $x$ .

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Step 1.4.2.1

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

$\frac{\frac{x \cdot x^{5}}{2 x} \frac{d}{d x} [2 x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.2.2

Cancel the common factors.

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Step 1.4.2.2.1

Factor $x$ out of $2 x$ .

$\frac{\frac{x \cdot x^{5}}{x \cdot 2} \frac{d}{d x} [2 x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.2.2.2

Cancel the common factor.

$\frac{\frac{x \cdot x^{5}}{x \cdot 2} \frac{d}{d x} [2 x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.2.2.3

Rewrite the expression.

$\frac{\frac{x^{5}}{2} \frac{d}{d x} [2 x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.3

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

$\frac{\frac{x^{5}}{2} (2 \frac{d}{d x} [x]) - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.4

Simplify terms.

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Step 1.4.4.1

Combine $2$ and $\frac{x^{5}}{2}$ .

$\frac{\frac{2 x^{5}}{2} \frac{d}{d x} [x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.4.2

Cancel the common factor of $2$ .

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Step 1.4.4.2.1

Cancel the common factor.

$\frac{\frac{2 x^{5}}{2} \frac{d}{d x} [x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.4.2.2

Divide $x^{5}$ by $1$ .

$\frac{x^{5} \frac{d}{d x} [x] - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.5

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

$\frac{x^{5} \cdot 1 - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.6

Multiply $x^{5}$ by $1$ .

$\frac{x^{5} - \ln (2 x) \frac{d}{d x} [x^{6}]}{x^{12}}$

Step 1.4.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{x^{5} - \ln (2 x) (6 x^{5})}{x^{12}}$

Step 1.4.8

Simplify with factoring out.

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Step 1.4.8.1

Multiply $6$ by $- 1$ .

$\frac{x^{5} - 6 \ln (2 x) x^{5}}{x^{12}}$

Step 1.4.8.2

Factor $x^{5}$ out of $x^{5} - 6 \ln (2 x) x^{5}$ .

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Step 1.4.8.2.1

Multiply by $1$ .

$\frac{x^{5} \cdot 1 - 6 \ln (2 x) x^{5}}{x^{12}}$

Step 1.4.8.2.2

Factor $x^{5}$ out of $- 6 \ln (2 x) x^{5}$ .

$\frac{x^{5} \cdot 1 + x^{5} (- 6 \ln (2 x))}{x^{12}}$

Step 1.4.8.2.3

Factor $x^{5}$ out of $x^{5} \cdot 1 + x^{5} (- 6 \ln (2 x))$ .

$\frac{x^{5} (1 - 6 \ln (2 x))}{x^{12}}$

Step 1.5

Cancel the common factors.

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Step 1.5.1

Factor $x^{5}$ out of $x^{12}$ .

$\frac{x^{5} (1 - 6 \ln (2 x))}{x^{5} x^{7}}$

Step 1.5.2

Cancel the common factor.

$\frac{x^{5} (1 - 6 \ln (2 x))}{x^{5} x^{7}}$

Step 1.5.3

Rewrite the expression.

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

Step 1.6

Simplify each term.

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Step 1.6.1

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

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

Step 1.6.2

Apply the product rule to $2 x$ .

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

Step 1.6.3

Raise $2$ to the power of $6$ .

$f' (x) = \frac{1 - \ln (64 x^{6})}{x^{7}}$

Step 2

Find the second derivative.

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Step 2.1

$\frac{x^{7} \frac{d}{d x} [1 - \ln (64 x^{6})] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{{(x^{7})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${(x^{7})}^{2}$ .

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Step 2.2.1.1

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

$\frac{x^{7} \frac{d}{d x} [1 - \ln (64 x^{6})] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{7 \cdot 2}}$

Step 2.2.1.2

Multiply $7$ by $2$ .

$\frac{x^{7} \frac{d}{d x} [1 - \ln (64 x^{6})] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.2.2

By the Sum Rule, the derivative of $1 - \ln (64 x^{6})$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \ln (64 x^{6})]$ .

$\frac{x^{7} (\frac{d}{d x} [1] + \frac{d}{d x} [- \ln (64 x^{6})]) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{7} (0 + \frac{d}{d x} [- \ln (64 x^{6})]) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.2.4

Add $0$ and $\frac{d}{d x} [- \ln (64 x^{6})]$ .

$\frac{x^{7} \frac{d}{d x} [- \ln (64 x^{6})] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (64 x^{6})$ with respect to $x$ is $- \frac{d}{d x} [\ln (64 x^{6})]$ .

$\frac{x^{7} (- \frac{d}{d x} [\ln (64 x^{6})]) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.3

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 $g (x) = 64 x^{6}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $64 x^{6}$ .

$\frac{x^{7} (- (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [64 x^{6}])) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.3.2

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

$\frac{x^{7} (- (\frac{1}{u} \frac{d}{d x} [64 x^{6}])) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.3.3

Replace all occurrences of $u$ with $64 x^{6}$ .

$\frac{x^{7} (- (\frac{1}{64 x^{6}} \frac{d}{d x} [64 x^{6}])) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4

Differentiate.

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Step 2.4.1

Combine $x^{7}$ and $\frac{1}{64 x^{6}}$ .

$\frac{- \frac{x^{7}}{64 x^{6}} \frac{d}{d x} [64 x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.2

Cancel the common factor of $x^{7}$ and $x^{6}$ .

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Step 2.4.2.1

Factor $x^{6}$ out of $x^{7}$ .

$\frac{- \frac{x^{6} x}{64 x^{6}} \frac{d}{d x} [64 x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

Factor $x^{6}$ out of $64 x^{6}$ .

$\frac{- \frac{x^{6} x}{x^{6} \cdot 64} \frac{d}{d x} [64 x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.2.2.2

Cancel the common factor.

$\frac{- \frac{x^{6} x}{x^{6} \cdot 64} \frac{d}{d x} [64 x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.2.2.3

Rewrite the expression.

$\frac{- \frac{x}{64} \frac{d}{d x} [64 x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.3

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{6}$ with respect to $x$ is $64 \frac{d}{d x} [x^{6}]$ .

$\frac{- \frac{x}{64} (64 \frac{d}{d x} [x^{6}]) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4

Simplify terms.

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Step 2.4.4.1

Multiply $64$ by $- 1$ .

$\frac{- 64 \frac{x}{64} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.2

Combine $- 64$ and $\frac{x}{64}$ .

$\frac{\frac{- 64 x}{64} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.3

Cancel the common factor of $- 64$ and $64$ .

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Step 2.4.4.3.1

Factor $64$ out of $- 64 x$ .

$\frac{\frac{64 (- x)}{64} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.3.2

Cancel the common factors.

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Step 2.4.4.3.2.1

Factor $64$ out of $64$ .

$\frac{\frac{64 (- x)}{64 (1)} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.3.2.2

Cancel the common factor.

$\frac{\frac{64 (- x)}{64 \cdot 1} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.3.2.3

Rewrite the expression.

$\frac{\frac{- x}{1} \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.4.3.2.4

Divide $- x$ by $1$ .

$\frac{- x \frac{d}{d x} [x^{6}] - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{- x (6 x^{5}) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.4.6

Multiply $6$ by $- 1$ .

$\frac{- 6 x \cdot x^{5} - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.5

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 2.5.1

Move $x^{5}$ .

$\frac{- 6 (x^{5} x) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.5.2

Multiply $x^{5}$ by $x$ .

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Step 2.5.2.1

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

$\frac{- 6 (x^{5} x^{1}) - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.5.2.2

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

$\frac{- 6 x^{5 + 1} - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.5.3

Add $5$ and $1$ .

$\frac{- 6 x^{6} - (1 - \ln (64 x^{6})) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{- 6 x^{6} - (1 - \ln (64 x^{6})) (7 x^{6})}{x^{14}}$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Multiply $7$ by $- 1$ .

$\frac{- 6 x^{6} - 7 (1 - \ln (64 x^{6})) x^{6}}{x^{14}}$

Step 2.7.2

Factor $x^{6}$ out of $- 6 x^{6} - 7 (1 - \ln (64 x^{6})) x^{6}$ .

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Step 2.7.2.1

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

$\frac{x^{6} \cdot - 6 - 7 (1 - \ln (64 x^{6})) x^{6}}{x^{14}}$

Step 2.7.2.2

Factor $x^{6}$ out of $- 7 (1 - \ln (64 x^{6})) x^{6}$ .

$\frac{x^{6} \cdot - 6 + x^{6} (- 7 (1 - \ln (64 x^{6})))}{x^{14}}$

Step 2.7.2.3

Factor $x^{6}$ out of $x^{6} \cdot - 6 + x^{6} (- 7 (1 - \ln (64 x^{6})))$ .

$\frac{x^{6} (- 6 - 7 (1 - \ln (64 x^{6})))}{x^{14}}$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $x^{6}$ out of $x^{14}$ .

$\frac{x^{6} (- 6 - 7 (1 - \ln (64 x^{6})))}{x^{6} x^{8}}$

Step 2.8.2

Cancel the common factor.

$\frac{x^{6} (- 6 - 7 (1 - \ln (64 x^{6})))}{x^{6} x^{8}}$

Step 2.8.3

Rewrite the expression.

$\frac{- 6 - 7 (1 - \ln (64 x^{6}))}{x^{8}}$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$\frac{- 6 - 7 \cdot 1 - 7 (- \ln (64 x^{6}))}{x^{8}}$

Step 2.9.2

Simplify the numerator.

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Step 2.9.2.1

Simplify each term.

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Step 2.9.2.1.1

Multiply $- 7$ by $1$ .

$\frac{- 6 - 7 - 7 (- \ln (64 x^{6}))}{x^{8}}$

Step 2.9.2.1.2

Multiply $- 7 (- \ln (64 x^{6}))$ .

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Step 2.9.2.1.2.1

Multiply $- 1$ by $- 7$ .

$\frac{- 6 - 7 + 7 \ln (64 x^{6})}{x^{8}}$

Step 2.9.2.1.2.2

Simplify $7 \ln (64 x^{6})$ by moving $7$ inside the logarithm.

$\frac{- 6 - 7 + \ln ({(64 x^{6})}^{7})}{x^{8}}$

Step 2.9.2.1.3

Apply the product rule to $64 x^{6}$ .

$\frac{- 6 - 7 + \ln (64^{7} {(x^{6})}^{7})}{x^{8}}$

Step 2.9.2.1.4

Raise $64$ to the power of $7$ .

$\frac{- 6 - 7 + \ln (4398046511104 {(x^{6})}^{7})}{x^{8}}$

Step 2.9.2.1.5

Multiply the exponents in ${(x^{6})}^{7}$ .

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Step 2.9.2.1.5.1

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

$\frac{- 6 - 7 + \ln (4398046511104 x^{6 \cdot 7})}{x^{8}}$

Step 2.9.2.1.5.2

Multiply $6$ by $7$ .

$\frac{- 6 - 7 + \ln (4398046511104 x^{42})}{x^{8}}$

Step 2.9.2.2

Subtract $7$ from $- 6$ .

$f'' (x) = \frac{- 13 + \ln (4398046511104 x^{42})}{x^{8}}$