Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 4.3

Combine fractions.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 5

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

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

Step 6

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

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