Chapter 3 Differentiation
Section 3.2
Differentiable
We say f ‘(x) is differentiable on (a,b) if f ‘(x)
exists for all x in (a,b).
Example
Prove f ( x)  x  12 x is differentiable.
Compute f ‘(x) and find the equation of the
tangent line at x = -3.
3
Derivative Notations
(expanded)
Standard notations for the derivative:
df dy
f '( x), y ', ,
dx dx
If we want a derivative at a specific point,
say x = 4, we write:
df dy
f '(4), y '(4),
,
dx 4 dx 4
Derivative Rules
#1. Derivatives of Constant Functions
If f(x)=b is a constant function, then
f ’(a)=0 for all a. (The derivative of any
constant is 0.)
Ex. Given f(x) = 5, f’(x) = 0
Derivative Rules
#2. Constant Multiple Rule
For any constant c, cf is differentiable and
(cf ) '  cf '
(A multiplicative constant stays the same
when a derivative is taken.)
Ex. The derivative of y = 4x2 is y’ = 4(y’(x2))
Derivative Rules (cont’d)
#3. Sum and Difference Rules
Assume that f and g are differentiable,
then f + g and f – g are differentiable and
( f  g )'  f ' g ' and ( f  g )'  f ' g '
(The derivative of a sum is the sum of the
derivative and the derivative of a difference is
the difference of the derivatives.)
Ex. Let f(x) = 2x – 4, then f’(x) = f’(2x) – f’(4) =
f’(2x) – 0
Derivative Rules (cont’d)
#4. The Power Rule
For all exponents n,
d n
n 1
x  nx
dx
Ex. If y = 7x4, then y’ = 28x3
The derivative of x-3/5 = -3/5x-8/5
Example
Find the points on the graph of
f (t )  t 3  12t  4 where the tangent line is
horizontal.
Example
Calculate
dg
dt
t 1
where
g (t )  t 3  2 t  t 4/5
Derivative Practice
Find the derivative of:
1.
7
y 8
2x
2. f ( x) 
1
7 3 x2
3. y  9 x6  5x4  2 x3  7
Matching
In the left hand column are
graphs of several functions.
In the right-hand column –
in a different order – are
graphs of the associated
derivative functions. Match
each function with its
derivative. [Note: The
scales on the graphs are
not all the same.]
Derivative Rules (cont’d)
#5. Derivative of ex
d x
e  ex
dx
Ex. If y = 2ex, then y’ = 2ex
#6. Derivative of bx (b is any base where b > 0)
d x
x
b  (ln b)(b ) Ex. If y = 3x, then y’ = ln3(3x)
dx
Example
Find the tangent line to the graph of
f ( x)  3e  5x at x  2.
x
2
Differentiability Implies
Continuity
Theorem
If f is differentiable at x=c, then f is
continuous at c.
Note: The converse is not always true.
Derivatives
Derivatives can’t be found at the
following points on a graph:
 Corners/Cusps
 Vertical Tangents
 Ever-Increasing Oscillations
Example
Show that f ( x)  x is continuous but not
differentiable at x=0.