Calculus Examples

$f (x) = \ln (3 x^{4})$

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 $g (x) = 3 x^{4}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{4}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x^{4}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [3 x^{4}]$

Step 1.3

Replace all occurrences of $u$ with $3 x^{4}$ .

$\frac{1}{3 x^{4}} \frac{d}{d x} [3 x^{4}]$

Step 2

Differentiate.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$\frac{1}{3 x^{4}} (3 \frac{d}{d x} [x^{4}])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $3$ and $\frac{1}{3 x^{4}}$ .

$\frac{3}{3 x^{4}} \frac{d}{d x} [x^{4}]$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{3}{3 x^{4}} \frac{d}{d x} [x^{4}]$

Step 2.2.2.2

Rewrite the expression.

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

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{x^{4}} (4 x^{3})$

Step 2.4

Simplify terms.

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Step 2.4.1

Combine $4$ and $\frac{1}{x^{4}}$ .

$\frac{4}{x^{4}} x^{3}$

Step 2.4.2

Combine $\frac{4}{x^{4}}$ and $x^{3}$ .

$\frac{4 x^{3}}{x^{4}}$

Step 2.4.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 2.4.3.1

Factor $x^{3}$ out of $4 x^{3}$ .

$\frac{x^{3} \cdot 4}{x^{4}}$

Step 2.4.3.2

Cancel the common factors.

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Step 2.4.3.2.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x^{3} \cdot 4}{x^{3} x}$

Step 2.4.3.2.2

Cancel the common factor.

$\frac{x^{3} \cdot 4}{x^{3} x}$

Step 2.4.3.2.3

Rewrite the expression.

$\frac{4}{x}$