Calculus Examples

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

Step 1

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

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

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 $g (x) = \arctan (x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Combine $\frac{1}{\arctan (x)}$ and $x$ .

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

Step 4

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

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply $\ln (\arctan (x))$ by $1$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Combine terms.

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Step 6.2.1

Multiply $\arctan (x)$ by $1$ .

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

Step 6.2.2

To write $\ln (\arctan (x))$ as a fraction with a common denominator, multiply by $\frac{\arctan (x) + \arctan (x) x^{2}}{\arctan (x) + \arctan (x) x^{2}}$ .

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

Step 6.2.3

Combine the numerators over the common denominator.

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

Step 6.3

Reorder terms.

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