Calculus Examples

$y = \ln (x^{5})$

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 $g (x) = x^{5}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{5}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{5}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{5}]$

Step 1.3

Replace all occurrences of $u$ with $x^{5}$ .

$\frac{1}{x^{5}} \frac{d}{d x} [x^{5}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{x^{5}} (5 x^{4})$

Step 3

Combine terms.

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Step 3.1

Combine $5$ and $\frac{1}{x^{5}}$ .

$\frac{5}{x^{5}} x^{4}$

Step 3.2

Combine $\frac{5}{x^{5}}$ and $x^{4}$ .

$\frac{5 x^{4}}{x^{5}}$

Step 3.3

Cancel the common factor of $x^{4}$ and $x^{5}$ .

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Step 3.3.1

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

$\frac{x^{4} \cdot 5}{x^{5}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

$\frac{x^{4} \cdot 5}{x^{4} x}$

Step 3.3.2.2

Cancel the common factor.

$\frac{x^{4} \cdot 5}{x^{4} x}$

Step 3.3.2.3

Rewrite the expression.

$\frac{5}{x}$