Let f(x) be a function and assume that for each value of x, we can calculate the slope of the tangent to the graph y = f(x) at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f(x) and denote it by f ´ (x).
Hence, for a function y = f(x), we denote
Note that f(x+h) − f(x) is the change in y corresponding to the change h in x. Leibniz denoted these changes by Δy and Δx respectively. Thus
For this reason, we often write.
The symbol should be taken as a whole and not treated as “dy divided by dx”. We read it as “dee-y by dee-x”. For clarity, we sometimes write or for the derivative of f(x).
To compute the derivative of a function f(x) is to differentiate the function f with respect to x. The process of finding a derivative is called differentiation.
For a straight line determined by y = mx + c,
the tangent of the graph (line) at each point coincides with the line.
Thus the tangent has a constant slope m. That is, .
We verify this by using the definition of the derivative:
Let f(x) = mx + c. Then
We find the derivative of f(x) = x2.
Find the derivative of these functions using the definition of the derivative.