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3.1 Derivatives of a Function, p. 98
AP Calculus AB/BC
0
limh
f a h f a
h
This limit is also called the derivative of f at a.
We write: 0
lim
h
f x h f xf x
h
“The derivative of f with respect to x is …”
In section 2.4, the slope of the curve of y = f(x) was defined
at the point where x = a to be
Example 1: 2Findd
xdx
0
' lim
h
f x h f xdf x f x
dx h
2 2
0lim
h
x h x
h
2 2 2
0
2lim
h
x xh h x
h
0
2lim
h
h x h
h
0lim 2
h
x h
2 x
0
Definition (Alternate) Derivative at a Point
The derivative of a function at a point x = a is:
' lim
x a
f x f af a
x a
provided that the limit exists.
' lim
x a
f x f af a
x a
Example 2: Use the alternate definition to differentiate
1, at 2.f x a
x
1 1
lim
x a
x ax a
lim
x a
a xax
x a
1lim
x a
a x
ax x a
1lim
x a ax 2
1
aFor a = 2:
2
1
2
1
4
−1
There are many ways to write the derivative of y f x
f x “f prime x” or “the derivative of f with respect to x”
y “y prime”
dy
dx“the derivative of y with respect to x”
df
dx“the derivative of f with respect to x”
df x
dx“the derivative of f of x”
( of of )d dx f x
dx does not mean d times x !
dy does not mean d times y !
dy
dx does not mean !dy dx
(except when it is convenient to think of it as division.)
df
dxdoes not mean !df dx
(except when it is convenient to think of it as division.)
(except when it is convenient to treat it that way.)
df x
dxdoes not mean times !
d
dx f x
y f x
y f x
The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.
Example 3: Graph the derivative of the function f.
2 3.y x
2 2
0
3 3limh
x h xy
h
2 2 2
0
2limh
x xh h xy
h
2y x
0lim 2h
y x h
0
Graph y and y′ where
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.
