Section 3.3
Rules of Differentiation
Part 1:
Read section 3.3 and answer:
1.
2.
3.
Part 2:
AWWWWW! Now that we have spent a lot of time using the limit definition of
derivative, we will learn the shortcut rules for finding derivatives:
Rule 1: Derivative of a Constant Function
If f has the constant value f ( x)  c , then
df
d
 (c )  0
dx dx
f ( x) 
Example:

4
, what is f ( x) ?
Rule 2: General Power Rule
If n is a real number, then
d n
( x )  nx n 1
dx
Note that your book deals with n as a positive integer—in reality, n could be any real no.
Examples:
Find f '( x) if:
1.
f ( x)  x 4
2.
f ( x)  x
3.
f ( x) 
1
x2
Rule 3: Constant Multiple Rule
If u is a differentiable function of x, and c is a constant, then
d
du
(cu )  c
dx
dx
Example: y  5 x 2 , what is
dy
?
dx
Rule 4: Derivative Sum (or Difference) Rule
If u and v are differentiable functions of x, then their sum u+v is differentiable at every
point where u and v are both differentiable. At such points,
d
du dv
(u  v) 

dx
dx dx
Find
dy
2
if y  5 x3  4 x 
x
dx
Derivative of the Natural Exponential Function
d x
(e )  e x
dx
Higher Order Derivatives: We can take the derivative of a derivative which would be called the second
derivative. The notation for the second derivative would be :
Find the second derivative of f ( x)  4 x3  7 x 2  5x  6
Find the third derivative of the above function.
f ( x) 
d 2 y d  dy  dy
  
 y
dx 2 dx  dx  dx
We’ve already alluded to the idea that horizontal tangent lines are important. So in order to find where
tangent lines are horizontal, we would set the first derivative equal to 0 and solve.
Find the equation of the horizontal tangent line(s) to the curve y  x3  9 x
Other examples:
Find the equation of the tangent line to f ( x)  x 2  7 x  4 at x=3.
What is a “normal line”?
Find the equation of the normal line to f ( x) 
x at x=4.