Calculus Examples

$\ln (y) \cdot \ln (y)$

Step 1

Raise $\ln (y)$ to the power of $1$ .

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

Step 2

Raise $\ln (y)$ to the power of $1$ .

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

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4

Add $1$ and $1$ .

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

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = \ln (y)$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $\ln (y)$ .

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

Step 5.2

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

$2 u \frac{d}{d y} [\ln (y)]$

Step 5.3

Replace all occurrences of $u$ with $\ln (y)$ .

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

Step 6

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

$2 \ln (y) \frac{1}{y}$

Step 7

Combine fractions.

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Step 7.1

Combine $\frac{1}{y}$ and $2$ .

$\frac{2}{y} \ln (y)$

Step 7.2

Combine $\frac{2}{y}$ and $\ln (y)$ .

$\frac{2 \ln (y)}{y}$

Step 8

Simplify $2 \ln (y)$ by moving $2$ inside the logarithm.

$\frac{\ln (y^{2})}{y}$