Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

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

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

Step 3.3.2.2.2

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

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

Step 3.3.2.2.3

Cancel the common factor.

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

Step 3.3.2.2.4

Rewrite the expression.

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

Step 3.3.2.2.5

Divide $x$ by $1$ .

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

Step 3.3.3

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

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

Step 3.3.4

Reorder terms.

$2 x \ln (x) + x$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 2 x \ln (x) + x$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 x \ln (x) + x$