Calculus Examples

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

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

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

Step 2

The derivative of $\log (x)$ with respect to $x$ is $\frac{1}{x \ln (10)}$ .

$x^{2} \frac{1}{x \ln (10)} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $x^{2}$ and $\frac{1}{x \ln (10)}$ .

$\frac{x^{2}}{x \ln (10)} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3.2

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

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Step 3.2.1

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

$\frac{x \cdot x}{x \ln (10)} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $x$ out of $x \ln (10)$ .

$\frac{x \cdot x}{x (\ln (10))} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3.2.2.2

Cancel the common factor.

$\frac{x \cdot x}{x \ln (10)} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3.2.2.3

Rewrite the expression.

$\frac{x}{\ln (10)} + \log (x) \frac{d}{d x} [x^{2}]$

Step 3.3

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

$\frac{x}{\ln (10)} + \log (x) (2 x)$

Step 3.4

Reorder terms.

$2 x \log (x) + \frac{x}{\ln (10)}$