Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x^{3})$ .

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

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

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Step 2.1

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

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

Step 2.2

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

$x (\frac{1}{u} \frac{d}{d x} [x^{3}]) + \ln (x^{3}) \frac{d}{d x} [x]$

Step 2.3

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Raise $x$ to the power of $1$ .

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

Step 3.2.2

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

$\frac{x \cdot 1}{x^{3}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]$

Step 3.2.3

Cancel the common factors.

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Step 3.2.3.1

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

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

Step 3.2.3.2

Cancel the common factor.

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

Step 3.2.3.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Simplify terms.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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Step 3.4.3.1

Cancel the common factor.

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

Step 3.4.3.2

Divide $3$ by $1$ .

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

Step 3.5

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

$3 + \ln (x^{3}) \cdot 1$

Step 3.6

Multiply $\ln (x^{3})$ by $1$ .

$3 + \ln (x^{3})$