Calculus Examples

$y = \arctan (\ln (x))$

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $1$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.4.4

Factor $x$ out of $x (\ln^{2} (x)) + x \cdot 1$ .

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