Calculus Examples

$\ln ({(x^{4})}^{2})$

Step 1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 1.1

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

$\frac{d}{d x} [\ln (x^{4 \cdot 2})]$

Step 1.2

Multiply $4$ by $2$ .

$\frac{d}{d x} [\ln (x^{8})]$

Step 2

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 $g (x) = x^{8}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{8}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{8}]$

Step 2.2

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

$\frac{1}{u} \frac{d}{d x} [x^{8}]$

Step 2.3

Replace all occurrences of $u$ with $x^{8}$ .

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Step 2.3.1

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

$\frac{1}{x^{4 \cdot 2}} \frac{d}{d x} [x^{8}]$

Step 2.3.2

Multiply $4$ by $2$ .

$\frac{1}{x^{8}} \frac{d}{d x} [x^{8}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{1}{x^{8}} (8 x^{7})$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $8$ and $\frac{1}{x^{8}}$ .

$\frac{8}{x^{8}} x^{7}$

Step 3.2.2

Combine $\frac{8}{x^{8}}$ and $x^{7}$ .

$\frac{8 x^{7}}{x^{8}}$

Step 3.2.3

Cancel the common factor of $x^{7}$ and $x^{8}$ .

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Step 3.2.3.1

Factor $x^{7}$ out of $8 x^{7}$ .

$\frac{x^{7} \cdot 8}{x^{8}}$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $x^{7}$ out of $x^{8}$ .

$\frac{x^{7} \cdot 8}{x^{7} x}$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{x^{7} \cdot 8}{x^{7} x}$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{8}{x}$