Derivative of Arctangent with definition of derivative I can't find the derivative of arctangent with definition of derivative.
Here's my way:
$\displaystyle\lim_{\Delta x\to\ 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ is a definition of derivative with limit. If we define $f(x)=\arctan(x)$, then we get:
$$\left[\lim_{\Delta x\to\ 0} \frac{\arctan(x+\Delta x)-\arctan(x)}{\Delta x}\right]=\left[\lim_{\Delta x\to\ 0} \frac{\arctan\left(\dfrac{\Delta x}{1+x(x+\Delta x)}\right)}{\Delta x}\right].$$ But I don't know how to continue. Thanks.

$$\lim_{\Delta x\to\ 0} \frac{\arctan\left(\dfrac{\Delta x}{1+x(x+\Delta x)}\right)}{\Delta x}=\lim_{\Delta x\to\ 0} \frac{\arctan\left(\dfrac{\Delta x}{1+x^2}\right)}{\Delta x}$$
Now let's introduce the variable $u$ by $$\tan u=\dfrac{\Delta x}{1+x^2}$$
Then $$\Delta x=(1+x^2)\tan u$$and your limit is $$\lim_{u\to\ 0}\frac{\arctan(\tan u)}{(1+x^2)\tan u}=\lim_{u\to\ 0}\frac{u}{(1+x^2)\tan u}=\frac1{1+x^2}\frac1{\lim_{u\to\ 0}\frac{\tan u} u}=\frac1{1+x^2}$$
