Calculus Examples

$\ln (x^{11})$

Step 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 $g (x) = x^{11}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{11}$ .

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{11}]$

Step 1.3

Replace all occurrences of $u$ with $x^{11}$ .

$\frac{1}{x^{11}} \frac{d}{d x} [x^{11}]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 11$ .

$\frac{1}{x^{11}} (11 x^{10})$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $11$ and $\frac{1}{x^{11}}$ .

$\frac{11}{x^{11}} x^{10}$

Step 2.2.2

Combine $\frac{11}{x^{11}}$ and $x^{10}$ .

$\frac{11 x^{10}}{x^{11}}$

Step 2.2.3

Cancel the common factor of $x^{10}$ and $x^{11}$ .

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Step 2.2.3.1

Factor $x^{10}$ out of $11 x^{10}$ .

$\frac{x^{10} \cdot 11}{x^{11}}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $x^{10}$ out of $x^{11}$ .

$\frac{x^{10} \cdot 11}{x^{10} x}$

Step 2.2.3.2.2

Cancel the common factor.

$\frac{x^{10} \cdot 11}{x^{10} x}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{11}{x}$