5 ## THE DERIVATIVEThe rate of change of a function The slope of a tangent line to a curve The definition of the derivative The derivative of The equation of a tangent to a curve The derivative of CALCULUS IS APPLIED TO THINGS that do not change at a constant rate. Velocity due to gravity, births and deaths in a population, units of Now, since we consider The slope of a straight line is this number:
(Topic 8 of Precalculus.) A straight line has one and only one slope; one and only one rate of change. If
straight line graph that relates them indicates constant speed. 45 miles per hour, say -- at every moment of time. The slope of a tangent line to a curve Calculus however is concerned with rates of change that are not constant. If this curve represents distance A secant to a curve
A tangent is a straight line that just touches a curve. A secant is a straight line that cuts a curve. Hence, consider the secant line that cuts the curve at points
But once again, the question calculus asks is: How is the function changing exactly at What is the slope of the We cannot however evaluate Therefore we will consider shorter and shorter distances Δ -- a sequence of slopes. And we will That slope, that limit, will be the value of what we will call the derivative. The difference quotient Let But when the value of Then Here, then, is the definition of the slope of the tangent line at The slope of the tangent line at Since Δ -- is called the Newton quotient, or the difference quotient. Calculating and simplifying it is a fundamental task in differential calculus. Again, the difference quotient is a function of Δ
The difference quotient then becomes: We now express the definition of the derivative as follows.
DEFINITION 5. By the derivative of a function We call that limit the function
And so we take the limit of the difference quotient as As for In practice, we have to simplify the difference quotient before letting
-- in such a way that we can divide it by To sum up: The derivative is a function -- a rule -- that assigns to each value of As an example, we will apply the definition to prove that the slope of the tangent to the function
In going from line 1) to line 2), we squared the binomial In going to line 3), we subtracted the In going to line 4), we divided the numerator by We can do that because We now complete the definition of the derivative and take the limit:
This is what we wanted to prove.
Whenever we apply the definition, we have to algebraically manipulate the difference quotient so that we can simply replace
Problem. Let a) at
Since b) at c) at Differentiable at According to the definition, a function will be differentiable at Where it does not have a tangent line Above are two examples. The function on the left does not have a derivative at As for the graph on the right, it is the absolute value function, The absolute value function nevertheless is continuous at (Conversely, though, if a function is differentiable at a point -- if there is a tangent -- it will also be continuous there. The graph will be smooth and have no break.) Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all values of their domains. Such functions are called differentiable functions. Can you name an elementary class of differentiable functions? To see the answer, pass your mouse over the colored area. Polynomials. Notations for the derivative Since the derivative is this limit: then the symbol for the limit itself is (Read: "dee- For example, if
"Dee- We also write
"
For example,
And so on. A simple difference quotient The difference quotient is a version of . And at times we will use the latter. That is, the change in the value of a function At times it will be convenient to express the difference quotient as
The student should now do Problems that require the definition of the derivative. Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |