Lecture Note Differentiation
() 0
()(
lim
xPx x Px
Px
Δ→ x⎡ +Δ − )⎤
′ = ⎢ ⎥
⎣ Δ ⎦
then
()
() () ( )
()
()
(^22)
0
2
0
0
400 6, 800 12, 000 400 68, 00 12, 000
lim400 800 6, 800
limlim 400 800 6, 800 800 6, 800xxxxx xx x x
Px
x
xxx x
x
xx xΔ→Δ→Δ→−+Δ+ +Δ− −−+ −
′ =
Δ
−Δ−Δ+ Δ
=
Δ
=−Δ−+ =−+
To find the value of x for which the slope of the tangent is zero, set the derivative
equal to zero and solve the resulting equation for x as follows:
Px′( )= 0then
800 6800 0
800 6800
6800
8.5
800
x
xx− +=
=
==
It follows that x = 8.5 are the x coordinates of the peak of the graph and that the
optimal selling price is $ 8.50 per radio.
2 Techniques of Differentiation
2.1 The Power Rule
For any number n,
()d xnnnx 1
dx= −
That is, to find the derivative ofxn, reduce the power of x by 1 and multiply by the
original power.
Example 1
Differentiate (find the derivative of) each of the following functions:
a. yx=^27 , b. 27
1
y
x= , c. yx= , d.1
y
x=
Solution
In each case, use exponential to write the function as a power function and then apply
the general rule.
a. ()^2727 27 1 27 26
d
x xx
dx
==−
b. 27 ()^27 27 1^28
1
27 27
dd
x xx
dx x dx⎛⎞==−=−−−−
⎜⎟
⎝⎠
−
c. () ()
(^111)
1/ 2^1122
222
dd
xxxx
dx dx
1
x−−
====d.
11 3 1
22 2
3111
(^222)
dd
xx x
dx x dx
1
x