Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (x^{2})$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\ln (x^{2})$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

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

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{\ln (x^{2})}$ .

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

Step 3.2

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

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

Step 3.3

Simplify terms.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

Factor $x$ out of $2 x$ .

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

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $x$ out of $x^{2} \ln (x^{2})$ .

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

Step 3.3.3.2.2

Cancel the common factor.

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

Step 3.3.3.2.3

Rewrite the expression.

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

Step 4

Simplify.

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Step 4.1

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

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

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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