Calculus Examples

$y = \ln (2 x^{3})$

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 $g (x) = 2 x^{3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 x^{3}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x^{3}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [2 x^{3}]$

Step 1.3

Replace all occurrences of $u$ with $2 x^{3}$ .

$\frac{1}{2 x^{3}} \frac{d}{d x} [2 x^{3}]$

Step 2

Differentiate.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{2 x^{3}} (2 \frac{d}{d x} [x^{3}])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $2$ and $\frac{1}{2 x^{3}}$ .

$\frac{2}{2 x^{3}} \frac{d}{d x} [x^{3}]$

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{2}{2 x^{3}} \frac{d}{d x} [x^{3}]$

Step 2.2.2.2

Rewrite the expression.

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

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{x^{3}} (3 x^{2})$

Step 2.4

Simplify terms.

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Step 2.4.1

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

$\frac{3}{x^{3}} x^{2}$

Step 2.4.2

Combine $\frac{3}{x^{3}}$ and $x^{2}$ .

$\frac{3 x^{2}}{x^{3}}$

Step 2.4.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.4.3.1

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} \cdot 3}{x^{3}}$

Step 2.4.3.2

Cancel the common factors.

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Step 2.4.3.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x^{2} \cdot 3}{x^{2} x}$

Step 2.4.3.2.2

Cancel the common factor.

$\frac{x^{2} \cdot 3}{x^{2} x}$

Step 2.4.3.2.3

Rewrite the expression.

$\frac{3}{x}$