Basic Differentiation Rules
Derivative Rules
• Theorem. [The Constant Rule] If k is a real
number such that f  x   k for all x in some open
interval I, then f '  x   0 for all x  I .
• Theorem. [The Power Rule] Let r be a rational
number, and let f  x   x r. Then
f '  x   rx
for all values of x where this expression is defined.
r 1
Examples
• Find derivatives for the following functions:
f  x   x100
g  x   13
h x  5 x
• Find the equation of the line tangent to the graph
of y  x3 at the point  2,8.
More Derivative Rules
• Theorem [The Constant Multiple Rule] Let k
represent a real number, and let f be a
differentiable function. Then the function kf is
also differentiable and
d
 kf  x    kf '  x  .
dx
• Example. Find the derivative of f  x   6 x3 .
•
Theorem [The Sum and Difference Rules] Let f and g be
differentiable functions. Then
d
 f  x   g  x    f '  x   g '  x 
dx
and
d
 f  x   g  x    f '  x   g '  x .
dx
•
Example. Find the derivative of each function.
f  x   6 x3  4 x
•
g  x   7 x3  8 x 2  7 x  2
Note. This theorem generalizes to any finite sum or difference.
Theorem.
d
sin x  cos x
dx
d
cos x   sin x
dx
Example. Find all values of x where the line tangent
to the graph of y  sin x has slope –1.
The Derivative As a Rate of
Change
• Slope.
f  x  f c
dy
 lim
dx xc
xc
f  x  f c

xc
y

x
rise

run
 rate of change in y with respect to x
• Velocity. Let s  t  be a function giving the position of
a point moving on a number line at time t.
s t   s c 
s '  c   lim
t 0
t c
s t   s c 

t c
distance

time
 rate of change in position
The derivative s '  c  gives the instantaneous velocity at time t  c.
The Derivative an Instantaneous Rate of Change
f  x  f c
dy
 lim
dx xc
xc
 instantaneous rate of change in y with respect to x
Example. A stone dropped from a bridge falls 16t 2 in
t seconds. Find the velocity after 3 seconds. If a
river flows 256 feet below the bridge, with what
velocity does the rock enter the water?