Explanation
The first derivative of a function f describes the instantaneous rate of change of f and is denoted by f'. The derivative f' indicates how steep the graph of f is at a given point.
The slope between two points P(x_1\,|\,y_1) and P(x_2\,|\,y_2) is calculated with \frac{y_2 - y_1}{x_2 - x_1}. Since f' is the slope of the graph of f at one (and not two) points, the distance between the points in the formula is allowed to approach zero. This gives us the definition of the derivative f':
Procedure
To determine the instantaneous rate of change, you differentiate the function using the differentiation rules. The most important differentiation rules are:
1. Power rule: f(x) = x^n \Rightarrow f'(x) = n \cdot x^{n-1}
2. Constant factor rule: f(x) = c \cdot g(x) \Rightarrow f'(x) = c \cdot g'(x)
3. Sum rule: f(x) = g(x) + h(x) \Rightarrow f'(x) = g'(x) + h'(x)
Examples
Differentiate the function f(x) = 4x^3.
(Power rule applied)
Differentiate the function f(x) = 5x^2 - 3x + 7.
(Power and sum rule applied)
Note
- ★The first derivative describes the instantaneous rate of change and the slope of a function.
- ★Use differentiation rules (power, constant factor, and sum rule) to differentiate graphs.
Exercises
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