DO NOW: Find the equation of the line tangent to
the curve f(x) = 3x2 + 4x at x = -2
3.1 – DERIVATIVE OF A FUNCTION
Remember… The slope of the line tangent to a curve
with equation y = f(x) at x = a:
Similarly, the velocity of an object with position function s = f(t) at t = a:
hafhafm
h
)()(lim0
hafhafav
h
)()()( lim0
The DerivativeDEFINITION The derivative of a function f at a
number x=a, denoted by f’(a), is
If this limit exists.
Also written:
hafhafaf
h
)()()(' lim0
axafxfaf
ax
)()()(' lim
Example 1 Find the derivative of the function
f(x) = x2 – 8x + 9 at the number a.
Interpretation as Slope of Tangent The tangent line to y = f(x) at (a, f(a)) is
the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.
The next two slides illustrate this interpretation of the derivative:
Slope of Tangent (2 interpretations)
Tangent Line Through a Point Using the point-slope form of the
equation of a line, we can write an equation of the line tangent to the curve y = f(x) at point (a, f(a)):
y – f(a) = f’(a)(x – a)
Example 2 Find an equation of the line tangent to
f(x) = x2 – 8x + 9 at the point (3,-6)
Interpretation as Rate of Change
The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.
Note that when the derivative is… Large, the y-values change
rapidly; Small, the y-values change
slowly.
Velocity and Speed Position Function: s = f(t)
Position along straight line
Velocity Function: f’(a) Velocity of the function at t = a
Speed: |v(t)|
Example 3 The position of a particle is given by the
equation of motion s = f(t) = 1/(1 + t), where t is measured in seconds and s in meters.
Find the velocity and speed after 2 seconds.
Derivative So far we have considered the derivative
of a function f at a fixed number a:
Now we change our point of view and let the number a vary:
hafhafaf
h
)()()(' lim0
hxfhxfxf
h
)()()(' lim0
Notation
y’“y prime” Nice and
brief, but does not name the independent variable
“dy dx” or “the derivative of y with respect to x”
Names both variables and uses d for derivative.
“df dx” of “the derivative of f with respect to x”
Emphasizes the function’s name
“d dx of f at x” or “the derivative of f at x;
Emphasizes the idea that differentiation is an operation performed on f.
dxdy
dxdf
)(xfdxd
Example 4 At right is the graph of a function f. Use
this graph to sketch the graph of the derivative f’(x).
Example 4 (solution)
Example 5 For the function f(x) = x3 – x
Find a formula for f’(x) Compare the graphs of f and f’
Example 5 (solution)
One-Sided Derivatives A function y = f(x) is differentiable on a
closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits[the right-hand derivative at a]
[the left-hand derivative at a]
Exist at the endpoints.
hafhaf
h
)()(lim0
hbfhbf
h
)()(lim0
Example 6 Show that the following function has left-
hand and right-hand derivatives at x=0, but no derivative there.
,2,2
xx
y 00
xx
Example 6 (solution)
Practice: Pg. 205 #1-19odd