Here we study the derivative of a function, as a function, in its own right.
The derivative of a function, as a function
We know that to find the derivative of a function at a point we write However, if we replace the given number with a variable , we now have This tells us the instantaneous rate of change at any given point .
The
notation:
means take the derivative of first, then evaluate at .
In other words, given a function of
Suppose you are given a polynomial and you are asked to compute . What is the
most efficient way to answer the question?
Compute directly from the definition of
derivative of a function at a point Compute the derivative as a function , and then evaluate at
derivative of a function at a point Compute the derivative as a function , and then evaluate at
Suppose you are given a polynomial and you are asked to compute , , and . What is
the most efficient way to answer the question?
Compute , , and directly from
the
definition of derivative of a function at a point Compute the derivative of as a function ,
and then evaluate at , , and
definition of derivative of a function at a point Compute the derivative of as a function ,
and then evaluate at , , and
Typically, a limit computation takes longer than evaluation, so when you need to
find that derivative of a function at multiple points, it is often quicker to compute
the derivative as a function, and then evaluate at the particular points.
Given a function from the real numbers to the real numbers, the derivative is also a function from the real numbers to the real numbers. Understanding the relationship between the functions and helps us understand any situation (real or imagined) involving changing values.
Let . What is ?
because is a number, and a number corresponds to a horizontal
line,
which has a slope of zero. because is a straight line with slope . We cannot solve this problem yet.
which has a slope of zero. because is a straight line with slope . We cannot solve this problem yet.
