Is the Derivative of Arctangent Equal to 1/x?

[dimension10]

According to Wolfram|Alpha, sometimes, the derivative of arctangent is

$$\frac{d}{dx}\arctan x =\frac{1}{1+{x}^{2}}$$

and sometimes,

$$\frac{d}{dx}\arctan x =\frac{\arctan x}{x}$$

So are both of them equal?

 

[asmani]

The second case is actually
$$\frac{d}{d\times x}\arctan (x)=\frac{\arctan (x)}{x}$$.

 

[HallsofIvy]

Okay, you have lost me. What does
$$\frac{d}{d\times x}$$
mean?

 

[asmani]

d over d multiplied by x, where d is a constant. Wolframalpha simplifies this to 1/x.

 

[dimension10]

Oh. So that must have happened the other time too...

 

[asmani]

Compare:

Notice the blank between d and x in the left one.
What input did you give in each time?

 

[HallsofIvy]

d over d multiplied by x, where d is a constant. Wolframalpha simplifies this to 1/x.

Thanks.