1. Interpretations of the Derivative Gottfried Wilhelm Leibniz
Lesson: ____
Section: 2.4
Interpretations of the Derivative
Alternative Notation (Leibniz):
𝑑 𝑑𝑥 (𝑦)
If y = f(x), we write or
to represent the derivative of y with respect to x.
Think of this as a tiny difference in y over a tiny difference in x. Thinking like this, it looks like rise over run now only with infinitesimally small differences.
Gottfried Wilhelm Leibniz

2. Given that profit is a function of time, P = f(t)
we speak of the derivative as f’(t) or
𝑑𝑃 𝑑𝑡
Advantage of Leibniz Notation?
We can clearly see the variable components of the rate
Determining units is easy!
Disadvantage of Leibniz Notation?
The notation for evaluating is awkward.

3. What does it mean to say that f’(2000) = 100?
The Cost of extracting T tons of copper ore from a mine is given by the function C = f(T) dollars.
What does it mean to say that f’(2000) = 100?

4. What does it mean to say that f’(2000) = 100?
The population of Mexico (in millions) is P = f(t) where t is the number of years since What do these mean? What are the units for each? f(2) = 125
f’(6) = 2
f-1(95.5) = 16
(f-1)’(95.5) = 0.46
The Cost of extracting T tons of copper ore from a mine is given by the function C = f(T) dollars.
What does it mean to say that f’(2000) = 100?

5. A wide variety of notations exist
Lagrange
f ʹ(x) ,  f ʺ(x)
Leibniz
The derivative of a function is written as…
ẋ , ẍ
Newton
Dx y  , D2f
Euler
Also, all of these mean the same thing!