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

Question

1491 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-04T03:41:03

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{2}{4 + x^{2}}$ $d/dxarctan(x/2)=2/(4+x^2)$

In general,

$\frac{d}{d x} arctan x = \frac{1}{1 + x^{2}}$ $d/dxarctanx=1/(1+x^2)$

So, this problem will also require application of the Chain Rule:

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{1}{1 + {(\frac{x}{2})}^{2}} \cdot \frac{d}{d x} (\frac{x}{2})$ $d/dxarctan(x/2)=1/(1+(x/2)^2)*d/dx(x/2)$

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{1}{1 + \frac{x^{2}}{4}} \cdot \frac{1}{2}$ $d/dxarctan(x/2)=1/(1+x^2/4)*1/2$

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{1}{2 + \frac{x^{2}}{2}}$ $d/dxarctan(x/2)=1/(2+x^2/2)$

We'll want to get rid of that fraction in the denominator. It doesn't look very good.

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{1}{\frac{4 + x^{2}}{2}}$ $d/dxarctan(x/2)=1/((4+x^2)/2)$

Finally,

$\frac{d}{d x} arctan (\frac{x}{2}) = \frac{2}{4 + x^{2}}$ $d/dxarctan(x/2)=2/(4+x^2)$

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