Simply put, a derivative of a function is the rate of change (slope) of a function at any given point. For example, the function f(x) = x has a constant slope of 1 when graphing it (y = x). The notation of a derivative uses the Prime symbol ('). So if f(x) = x then f'(x) = 1.
It's rather simple for constant-slope functions, since the derivative is simply equal to the constant slope. For a variable-slope function, such as x^2, it's slightly more difficult. However, there are tricks to learn to make calculating the derivative of a function much easier than actually using lim(f(x)) as x-> infinity (or something like that. I forget exactly what it is). For example, using the function f(x) = x^2, the slope of the graph of f(x) at any point x is 2x, that is f'(x) = 2x.
In general: if f(x) = ax^b then f'(x) = abx^(b-1)
So for another example: f(x) = 2x^3 then f'(x) = 6x^2 and f''(x) = 12x (second derivative; That is, the derivative of the derivative function).
And of course, if f(x) = C, then f'(x) = 0
If, however, you get other functions, such as f(x) = sin(x) or f(x) = ln(x), then there are other rules you need to apply.
Specifically, here are a couple of general rules:
if f(x) = sin(x) then f'(x) = cos(x)
if f(x) = cos(x) then f'(x) = -sin(x)
if f(x) = ln(x) then f'(x) = 1/x
And there are product rules, quotient rules, and some other kind of rules but I forget the names. In words:
The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.
Example: if f(x) = 2x * x^2 then f'(x) != 2 * 2x as you might initially think. Instead, it must be seperated into g(x) = 2x and h(x) = x^2 so:
f(x) = g(x) * h(x). So, applying the rule, we get
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= 2 * x^2 + 2x * 2x
= 6x^2
The derivative of a quotient is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared.
Example:
f(x) = 2x/x^2
f'(x) = (2(x^2) - 2x(2x)) / x^4
= -2x^2 / x^4
= -2 / x^2
And the other one, which I don't remember, is: The derivative of a function (like sin(x^2)) is the natural derivative of the function (like cos(x^2)) times the derivative of the inside (2x). So if f(x) = sin(x^2) then f'(x) = cos(x^2) * 2x. Or if f(x) = ln(x^2) then f'(x) = 2/x (skipping simplification steps)
If you need more help, this website may be able to help. It can solve integrals, which are the opposite of derivatives (so if you put in cos(x^2) * 2x it will give you sin(x^2)) and it used to be able to solve derivativesm but I tried it this morning, and it didn't seem to be working. In any case, it can help:
QuickMath
