Calculus Examples

$y = \ln^{12} (x) + \ln (x^{12})$

Step 1

Remove parentheses.

$y = \ln^{12} (x) + \ln (x^{12})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln^{12} (x) + \ln (x^{12}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $\ln^{12} (x) + \ln (x^{12})$ with respect to $x$ is $\frac{d}{d x} [\ln^{12} (x)] + \frac{d}{d x} [\ln (x^{12})]$ .

$\frac{d}{d x} [\ln^{12} (x)] + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2

Evaluate $\frac{d}{d x} [\ln^{12} (x)]$ .

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Step 4.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{12}$ and $g (x) = \ln (x)$ .

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Step 4.2.1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{12}] \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 12$ .

$12 {u_{1}}^{11} \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2.1.3

Replace all occurrences of $u_{1}$ with $\ln (x)$ .

$12 \ln^{11} (x) \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2.2

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

$12 \ln^{11} (x) \frac{1}{x} + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2.3

Combine $\frac{1}{x}$ and $12$ .

$\frac{12}{x} \ln^{11} (x) + \frac{d}{d x} [\ln (x^{12})]$

Step 4.2.4

Combine $\frac{12}{x}$ and $\ln^{11} (x)$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{d}{d x} [\ln (x^{12})]$

Step 4.3

Evaluate $\frac{d}{d x} [\ln (x^{12})]$ .

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Step 4.3.1

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 $g (x) = x^{12}$ .

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Step 4.3.1.1

To apply the Chain Rule, set $u_{2}$ as $x^{12}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [x^{12}]$

Step 4.3.1.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{1}{u_{2}} \frac{d}{d x} [x^{12}]$

Step 4.3.1.3

Replace all occurrences of $u_{2}$ with $x^{12}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{1}{x^{12}} \frac{d}{d x} [x^{12}]$

Step 4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 12$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{1}{x^{12}} (12 x^{11})$

Step 4.3.3

Combine $12$ and $\frac{1}{x^{12}}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{12}{x^{12}} x^{11}$

Step 4.3.4

Combine $\frac{12}{x^{12}}$ and $x^{11}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{12 x^{11}}{x^{12}}$

Step 4.3.5

Cancel the common factor of $x^{11}$ and $x^{12}$ .

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Step 4.3.5.1

Factor $x^{11}$ out of $12 x^{11}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{x^{11} \cdot 12}{x^{12}}$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $x^{11}$ out of $x^{12}$ .

$\frac{12 \ln^{11} (x)}{x} + \frac{x^{11} \cdot 12}{x^{11} x}$

Step 4.3.5.2.2

Cancel the common factor.

$\frac{12 \ln^{11} (x)}{x} + \frac{x^{11} \cdot 12}{x^{11} x}$

Step 4.3.5.2.3

Rewrite the expression.

$\frac{12 \ln^{11} (x)}{x} + \frac{12}{x}$

Step 5

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

$y' = \frac{12 \ln^{11} (x)}{x} + \frac{12}{x}$

Step 6

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

$\frac{d y}{d x} = \frac{12 \ln^{11} (x)}{x} + \frac{12}{x}$