Understanding steps to obtain derivative of $|x_n|^{\frac{3}{7}}$ I was trying to solve the following derivative 
$$|x_n|^{\frac{3}{7}}$$
as follows
$$(|x_n|^{\frac{3}{7}})' \\= \frac{3}{7}(|x_n|^{\frac{3}{7} - 1})  \cdot (|x_n|)' \\=  \frac{3}{7}|x_n|^{\frac{-4}{7}} \cdot \frac{x_n}{|x_n|} \\= \frac{3}{7}|x_n|^{\frac{-4}{7} + 1} \cdot x_n \\= \frac{3}{7}|x_n|^{\frac{3}{7}} \cdot x_n$$
which is different from the result that this online calculator gives me, which is
$$\frac{3}{7} x^{\frac{3}{7}} \frac{1}{|x|}$$
What did I do wrong?

The absolute value just implies you have symmetry about the y-axis. 
Assume $x>0$, then
$$\frac{d}{dx} x^{3/7} = \frac{3}{7}x^{-4/7} = \frac{3}{7}x^{3/7-1}=\frac{3}{7}\frac{x^{3/7}}{x}(x>0)$$
If $x<0$ then we just need to reverse the sign of the argument so $-x$ is positive:
$$\frac{d}{dx} (-x)^{3/7} = \frac{-3}{7}(-x)^{-4/7} = \frac{-3}{7}(-x)^{3/7-1}=\frac{-3}{7}\frac{(-x)^{3/7}}{-x}=\frac{3}{7}\frac{(-x)^{3/7}}{x}(x<0)$$
Note that the final result has the same exact form except that the numerator is corrected to always be positive, while the denominator is allowed to vary in sign. We can merge these two results by simply taking the numerator to be $|x|^{3/7}$ and we get the final form:
$$ \frac{3}{7}\frac{|x|^{3/7}}{x} \equiv \frac{3}{7}\frac{x}{|x|^{11/7}}$$
Note that the formula give by your online calculator is not correct: $\frac{3}{7}\frac{x^{3/7}}{|x|}$ is a complex number for $x<0$! (just calculate $(-2)^{3/7}$).
As for your derivation, you made a mistake in going from 
$$\frac{3}{7}|x_n|^{\frac{-4}{7}} \cdot \frac{x_n}{|x_n|}$$
to 
$$\frac{3}{7}|x_n|^{\frac{-4}{7} + 1} \cdot x_n$$
You distributed the numerator twice (note the +1 in exponent yet $x$ is still there) and lost the denominator.
