Answer:
derivative is a function which measures the slope. It depends upon x in some way, and is found by differentiating a function of the form y = f (x). When x is substituted into the derivative, the result is the slope of the original function y = f (x).
There are many different ways to indicate the operation of differentiation, also known as finding or taking the derivative. The choice of notation depends on the type of function being evaluated and upon personal preference.
Suppose you have a general function: y = f(x). All of the following notations can be read as "the derivative of y with respect to x" or less formally, "the derivative of the function."
f'(x) f' y' df/dx dy/dx d/dx [f(x)].
[HINT: don't read the last three terms as fractions, read them as an operation.
For example, read: " dx/dy = 3x"
As: "the function that gives the slope is equal to 3x"
Let's try some examples. Suppose we have the function : y = 4x3 + x2 + 3.
After applying the rules of differentiation, we end up with the following result:
dy/dx = 12x2 + 2x.
How do we interpret this? First, decide what part of the original function (y = 4x3 + x2 + 3) you are interested in. For example, suppose you would like to know the slope of y when the variable x takes on a value of 2. Substitute x = 2 into the function of the slope and solve:
dy/dx = 12 ( 2 )2 + 2 ( 2 ) = 48 + 4 = 52.
Therefore, we have found that when x = 2, the function y has a slope of + 52.
Now for the practical part. How do we actually determine the function of the slope? Almost all functions you will see in economics can be differentiated using a fairly short list of rules or formulas, which will be presented