Math 1314
Lesson 5
Section 11.4 - 12.5 Finding Derivatives
We are interested in finding the slope of the tangent
line at a specific point.
We need a way to find the slope of the tangent line
analytically for every problem that will be exact every
time.
We can draw a secant line across the curve, then
take the coordinates of the two points on the curve, P
and Q, and use the slope formula which give us the
slope of secant line.
f(x+h)
Now let’s write these points as ordered pairs, say
P(𝑥, 𝑓(𝑥)) and Q(𝑥 + ℎ, 𝑓(𝑥 + ℎ)). Then the slope
of the secant line is:
𝑦2 −𝑦1
𝑥2 −𝑥1
=
𝑓(𝑥+ℎ)−𝑓(𝑥)
(𝑥+ℎ)−𝑥
=
𝑓 (𝑥+ℎ)−𝑓(𝑥)
ℎ
Q
f(x)
.
P
x
x+h
Now suppose we move point Q closer to point P. In other words, ℎ approaches 0 (ℎ → 0).
Then we get the slope of the tangent line at (𝑥, 𝑓(𝑥)).
Therefore, the slope of the tangent line is lim
h 0
Slope of the secant line :
f ( x  h)  f ( x )
.
h
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
Slope of the tangent line at P( x, f ( x)) : lim
h 0
Section 11.4 – 12.5 Finding Derivatives
f ( x  h)  f ( x )
= 𝑓′(𝑥) if the limit exists.
h
1
Example 1: Find the derivative of 𝑓(𝑥) = 𝑥 2 using the definition.
Rules for Finding Derivatives
We can use the limit definition of the derivative to find the derivative of every function, but it
isn’t always convenient. Fortunately, there are some rules for finding derivatives which will
make this easier.
𝑓 ′ (𝑥) =
d
 f (x) means “the derivative of f with respect to x.”
dx
Rule 1: The Derivative of a Constant
d
c  c  0, where c is a constant.
dx
Example 2: Find the derivative of each function.
a. f ( x)  17
Section 11.4 – 12.5 Finding Derivatives
b. g ( x)  11
2
 
Rule 2: The Power Rule
d n
x  [ x n ]  nx n 1 for any real number n
dx
Example 3: Find the derivative of each function.
a. f ( x )  x
b. g ( x)  x
5
1
h
(
x
)

c.
x3
d. j ( x) 
10
x
Rule 3: Derivative of a Constant Multiple of a Function
d
cf ( x)  c d  f ( x) where c is any real number (which means [cf ( x)]  c[ f ( x)] ).
dx
dx
Example 4: Find the derivative of each function.
a.
c.
h( x )  5 x
g ( x) 
b.
f ( x)  6 x 4
1 4
x
4
Section 11.4 – 12.5 Finding Derivatives
3
Rule 4: The Sum/Difference Rule
d
 f ( x)  g ( x)  d  f ( x)  d g ( x)
dx
dx
dx
Example 5: Find the derivative:
f ( x)  10 x 4  3x 2  6 x  5.
1
7
6
f
(
x
)


2
x

6
x

 1.
Example 6: Find the derivative:
4
2x
Note, there are many other rules for finding derivatives “by hand.” We will not be using
those in this course. Instead, we will use GeoGebra for finding more complicated derivatives.
Example 7: Find the derivative of 𝑓(𝑥) = (2𝑥 + 1)3 − 4𝑥 + 𝑥𝑒 𝑥 and 𝑓 ′ (1).
Command:
Answer:
Command:
Answer:
Section 11.4 – 12.5 Finding Derivatives
4
Example 8: The median price of a home in one part of the US can be modeled by the function
P(t )  0.01363t 2  9.2637t  125.84 , where P(t) is given in thousands of dollars and t is
the number of years since the beginning of 1995. According to the model, at what rate were
median home prices changing at the beginning of 2005?
Command:
Section 11.4 – 12.5 Finding Derivatives
Answer:
5