So how would I be able to get this summed down to using my fingers?

Lulz I suck at maths.

I think (I may be very wrong, though), that both are equally interchangeable, at first you always use f'(x) when you start with derivatives because you're only working with one variable, when you switch to two or more variables you get accustomed to using df/dx and, at least me, always use the latter now, it tells more information than the first one (because it tells you in which variable you're derivating)

yeah, zexen, you're pretty much right.

clearly you're not inputing functions the right way for your program, lol.

absolute values are tricky for computers, because they don't get the derivative right.  you can use Sqrt(f(x)^2) instead.

anyway, so i just went ahead and checked in mathematica.

>> f[x_] = x*Sqrt[Sin[8x]^2]

and the derivative is basically the same thing i had--i expressed my solution in terms of domain, of course.

if you Simplify[...] that, you get:

(1+8x Cot(8x)) * Sqrt(sin(8x)^2)

where Cot is, of course, the cotangent function, i.e. cos(x)/sin(x).

which is exactly the same thing i had.

Yeah I must of typed it in wrong haha

it accepted (1+8xcot(8x))*sqrt(sin(8x)^2)

thanks

I know both notations but I just didn't think it made sense to differentiate f(x) into df/dx because 'f' isn't actually a variable, so writing what is essentially the notation for 'f derived in terms of x' doesn't make much sense.

it's really just a notation, and they're both used interchangeabily.  the f' notation tends to used more often when the independent variation is t, i.e. time.  but f' for f(x) is pretty ubiquitous too.  it varies in different fields of study so it's just a matter of preference and/or for historical reasons.

there're a bunch of notations for taking derivatives since it's pretty much the most basic mathetical operation and comes up in so many fields.  here are some of the ones i've come across at one point or another:

Newton's "dot" notation: see this a lot in mechanics (as in a branch of physics).  pretty much used only when you mean to take the time derivative.

Leibniz's dy/dx notation: probably the most used, largely because it is very closely related to the actual definition of a derivative.  it's also easily extended to partial derivatives, using the greek little delta symbol.

the f' prime notation: used when it's clear when the underlying independent variable is.  so perfect for single-variable functions or variables that depend on just 1 other variable.

the subscript notation: as in f_x .  usually used in partial derivatives. more of a shorthand for the more cumbersome Leibniz's notation.

using a big "D": usually used in context of thinking of taking the derivative as a mathematical operator.