Calculus Examples

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

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 $f (x) = x^{2}$ and $g (x) = \ln (2 x)$ .

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

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 $g (x) = 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $x$ out of $2 x$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Simplify terms.

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1

Cancel the common factor.

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

Step 3.4.2.2

Divide $x$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Multiply $x$ by $1$ .

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

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x + \ln (2 x) (2 x)$

Step 3.8

Reorder terms.

$2 x \ln (2 x) + x$