Calculus Examples

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

Step 1

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

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

Step 2

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 $g (x) = \ln (x)$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor.

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

Step 4.2.2.4

Rewrite the expression.

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

Step 4.2.2.5

Divide $x$ by $1$ .

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

Step 4.3

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $2$ .

$2 x + 4 \ln (x) x$

Step 5.3

Reorder terms.

$4 x \ln (x) + 2 x$