Calculus Examples

$\ln (x^{k})$

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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 $n = k$ .

$\frac{1}{x^{k}} (k x^{k - 1})$

Step 2.2

Simplify terms.

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Step 2.2.1

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

$\frac{k}{x^{k}} x^{k - 1}$

Step 2.2.2

Combine $\frac{k}{x^{k}}$ and $x^{k - 1}$ .

$\frac{k x^{k - 1}}{x^{k}}$

Step 2.2.3

Cancel the common factor of $x^{k - 1}$ and $x^{k}$ .

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Step 2.2.3.1

Factor $x^{k}$ out of $k x^{k - 1}$ .

$\frac{x^{k} (k x^{- 1})}{x^{k}}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Multiply by $1$ .

$\frac{x^{k} (k x^{- 1})}{x^{k} \cdot 1}$

Step 2.2.3.2.2

Cancel the common factor.

$\frac{x^{k} (k x^{- 1})}{x^{k} \cdot 1}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{k x^{- 1}}{1}$

Step 2.2.3.2.4

Divide $k x^{- 1}$ by $1$ .

$k x^{- 1}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$k \frac{1}{x}$

Step 3.2

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

$\frac{k}{x}$