What is the derivative of 1/2 arctan(x/2) ?

The derivative of 1/2 arctan(x/2) is 1/(x^2+4)

What is the first derivative of 1/2 arctan(x/2) ?

The first derivative of 1/2 arctan(x/2) is 1/(x^2+4)

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\frac{d}{d x} (\frac{1}{2} arctan (\frac{x}{2}))

\frac{1}{x ^{2} + 4}

Solution steps

\frac{d}{d x} (\frac{1}{2} arctan (\frac{x}{2}))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} \frac{d}{d x} (arctan (\frac{x}{2}))

Apply the chain rule:

\frac{1}{( \frac{x}{2} ) ^{2} + 1} \frac{d}{d x} (\frac{x}{2})

= \frac{1}{( \frac{x}{2} ) ^{2} + 1} \frac{d}{d x} (\frac{x}{2})

\frac{d}{d x} (\frac{x}{2}) = \frac{1}{2}

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

Simplify

\frac{1}{2} \cdot \frac{1}{( \frac{x}{2} ) ^{2} + 1} \cdot \frac{1}{2} : \frac{1}{x ^{2} + 4}

= \frac{1}{x ^{2} + 4}

What is the derivative of 1/2 arctan(x/2) ?
The derivative of 1/2 arctan(x/2) is 1/(x^2+4)
What is the first derivative of 1/2 arctan(x/2) ?
The first derivative of 1/2 arctan(x/2) is 1/(x^2+4)