Download Section 2.2 Student Notes Derivative as a function.

Section 2.2 The Derivative as a Function
f ' ( x ) = lim h→0
f ( x + h) − f ( x)
h
In the previous section we found the value of the derivative of f at a
particular point a. We learned that this value was equal to the slope of the
tangent line at the point (a, f(a)).
Now if we solve for x in general we derive a function that describes the slope
of f for any point in the domain of f’. The domain of the derivative may be
smaller than the original function.
Exercise 1. Use the given graph of f ( x ) to sketch the graph of f ' ( x ) .
Find the derivative of the function using the definition of the derivative.
Exercise 2.
f ( x ) = x State the domain of the function and the domain of
the derivative.
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Exercise 3. Find the derivative of the function using the definition of the
derivative. State the domain of the function and the domain of the derivative.
f ( x) =
2− x
x +1
Other Notations for the Derivative
f '( x) = y ' =
dy df
d
=
=
f ( x ) = Df ( x ) = Dx f ( x )
dx dx dx
The symbols D and
d
are called differentiation operators.
dx
Definition A function f is differentiable at a if f ' ( a ) exists.
Exercise 4. Consider where the function f ( x ) = x is differentiable.
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Theorem: If f is differentiable at a , then f is continuous at a.
Three ways f is not differentiable at a.
1________ ____________________ _______
2. __________________________________
3. ___________________________________
d2y
dny
(n)
(n)
Higher Derivatives: f '' ( x ) = y '' = 2
y = f ( x) =
dx
dx
When taking higher, nth order derivatives, the operator may also be written
dn
f ( x ) , D n or Dx n
dx
Exercise 4. If f ( x ) = x3 + x,
find f '' ( x )
Given s ( t ) as the position function, what do the first and second derivatives represent?
The figure shows the graphs of three functions. One is the position function of a car,
one is the velocity of the car, and one is its acceleration. Identify each curve.
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