Download Lec 4

Derivative
Recall:
First Principle Definition
The derivative of y = f (x) at x = a:
f (a + h) − f (a)
h→0
h
f 0 (a) = lim
Example:
Consider f (x) = x2 . Find the derivative and the equation of the tangent line
at x = 2.
f (2 + h) − f (2)
h→0
h
(2 + h)2 − 22
lim
h→0
h
2
2 + h2 + 4h − 22
lim
h→0
h
h2 + 4h
lim
h→0
h
h(h + 4)
lim
h→0
h
lim h + 4
f 0 (2) = lim
=
=
=
=
=
h→0
= 0+4=4
So the slope of the tangent line to f (x) = x2 at point (2, f (2)) = (2, 22 ) is
4. Hence, the equation of the tangent line is y = 4x + b. Substituting (2, 4)
and solving for b we get:
y = 4x − 4.
What is f 0 (x)?
The derivative of a function is itself a function of x and we can find it as such
f (x + h) − f (x)
.
h→0
h
f 0 (x) = lim
Examples:
Find f 0 (x) for each case.
1. f (x) = x
2. f (x) = x2
2
3. f (x) = x3
4. f (x) =
1
x
3
Observation?
5. If f (x) = mx + b
4
Example:
The position of a moving object is given by the position function s(t) =
2t2 + 3t + 2 (where s is in meters and t is in seconds). What is the velocity
function of this object? (Use the first principle formula.)
5
Does the derivative always exist? No!
f (a + h) − f (a)
exists.
h→0
h
f (x) is differentiable at x = a if f 0 (a) = lim
Examples:
1. f (x) =
x+2
x−2
at x = 2.
6
2. f (x) =
3x + 1, x ≤ −1
2 − x2 , x > −1
7
(
x
x≥0
3. f (x) = |x| =
.
−x x < 0
The derivative of f (x) does not exist if a is not in the domain or if f (x) is
not continuous at a or has a cusp at a or has a vertical tangent or etc.
8
Remark: Here is an alternative notation for the derivative. Suppose we
have f (x) = y
df
y 0 = f 0 (x) =
dx
and
df
f 0 (a) =
|x=a .
dx
9
Derivative of Polynomials
f (x) = a0 + a1 x + a2 x2 + a3 x3 + · · · + an xn
is the general form of a polynomial.
• The Constant Rule:
f (x) = c
• The Constant Multiple Rule:
f (x) = cg(x)
10
• Sum/Difference Rule
f (x) = p(x) ± q(x)
• The Power Rule:
f (x) = xn
Table 1: Derivative Rules
Constant Rule
f (x) = c
f 0 (x) = 0
n
0
Power Rule
f (x) = x
f (x) = nxn−1
Constant Multiple Rule
f (x) = cg(x)
f 0 (x) = cg 0 (x)
Sum/Difference Rule
f (x) = p(x) ± q(x) f 0 (x) = p0 (x) ± q 0 (x)
11
Examples:
Use the derivative rules to find the derivatives of the given functions.
1. f (x) = 5x6 − 4x3 + 6
2. f (x) =
3. y =
√
3
x
1
x
12
4. f (t) = 3t2 −
5.
√
t
−1
x5
Example:
Find the equation of the tangent line to the curve f (x) = 2x3 + 6x2 − 4x + 3
at x = −2.
13
Example:
The height above the ground, in meters, of an object dropped from the top
of a building of height 60 m, after t seconds, is h(t) = 60 − 4.9t2 .
(i) How fast is the object falling after 2 s ?
(ii) What is the speed of the object when it hits the ground ?
14
The Product Rule
((f (x)g(x))0 = f 0 (x)g(x) + g 0 (x)f (x)
or
d
(f (x)g(x)) = f 0 (x)g(x) + g 0 (x)f (x)
x
How do we prove this?
15
Examples:
1.
d
(x2 + 1)(x2 + 3x − 2) =
dx
2. f (t) = (t + 1)(3t4 − t2 ), find f 0 (t).
3. y = (2x2 + 1)2 = (2x2 + 1)(2x2 + 1), find y 0 .
16
Example:
A local coffee shop sells 100 mochas per day at a price of $2.75. They have
determined that for each 25 cent price increase, they will sell 5 fewer per day.
Write the revenue function. What is the derivative of this function?
17