Calculus Examples

$y \ln (y)$

Step 1

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

$y \frac{d}{d y} [\ln (y)] + \ln (y) \frac{d}{d y} [y]$

Step 2

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

$y \frac{1}{y} + \ln (y) \frac{d}{d y} [y]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $y$ and $\frac{1}{y}$ .

$\frac{y}{y} + \ln (y) \frac{d}{d y} [y]$

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Cancel the common factor.

$\frac{y}{y} + \ln (y) \frac{d}{d y} [y]$

Step 3.2.2

Rewrite the expression.

$1 + \ln (y) \frac{d}{d y} [y]$

Step 3.3

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

$1 + \ln (y) \cdot 1$

Step 3.4

Multiply $\ln (y)$ by $1$ .

$1 + \ln (y)$