What is the derivative of $y = arctan (\frac{x}{2})$ $y= arctan( x/2)$ ?

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Answer 1 · 2016-03-22T07:07:28

$\frac{d y}{d x} = \frac{2}{x^{2} + 4}$ $dy/dx=2/(x^2+4)$

$y = arctan (\frac{x}{2})$ $y = arctan(x/2)$

$\Rightarrow tan (y) = \frac{x}{2}$ $=> tan(y) = x/2$

$\Rightarrow \frac{d}{d x} tan (y) = \frac{d}{d x} \frac{x}{2}$ $=> d/dxtan(y) = d/dx x/2$

$\Rightarrow {sec}^{2} (y) \frac{d y}{d x} = \frac{1}{2}$ $=> sec^2(y)dy/dx = 1/2$

$:. dy/dx = 1/(2sec^2(y))$

$= 1/(2*(x^2+4)/4)$
(To see why, try drawing a right triangle with angle $y$ such that $tan(y) = x/2$ . What does $sec^2(y)$ equal?)

$=2/(x^2+4)$

What is the derivative of y= arctan( x/2)y=arctan(x2)y= arctan( x/2)?