I think I understand the general math approach, finally, suggested by MrAl in post #5. I would like to put down how I thought and hope that MrAl or other experts here could check and see if I got the correct concepts.
The objective is to prove that the operating point in Fig 5.15 is more sensitive to the base voltage Vbe at lower Vcc.
First I think I need to establish a
fixed output voltage Vc or Vce=Vce0, a fixed collector current Ic=Ic0 to meet some kind of requirements. This can be done by choosing suitable (Vcc, Rc, Rb) pairs, (Vcc1, Rc1, Rb1) for example, to make the transistor operated in active mode and satisfy
Rc1=(Vcc1-Vce0)/Ic0 (1)
and
Rb1=(Vcc1-Vbe0)/Ib0 (2)
in which Ib0=Ic0/beta and assumes that in such Vcc1, Rc1 and Rb1 we have base-emitter voltage Vbe0
Now look at the circuit below, which represents one particular experiment, and the equation
Ib=(Vcc-Vbe)/Rb (3)
If I substitute this particular Vcc1 and the corresponding Rb1 which is decided by equation (2) into (3), I get Ib=(Vcc1-Vbe)Ib0/(Vcc1-Vbe0).
The reason why Ib and Vbe are hold as variables is that I want to look at the variations in Ib while Vbe is changed in this experiment., so taking the derivative Ib with respect to Vbe, I get
dIb/dVbe=-Ib0/(Vcc1-Vbe0).
I can see that there's nothing special about Vcc1, it cab be any values among (Vcc, Rc, Rb) pairs, so I can write dIb/dVbe=-Ib0/(Vcc-Vbe0) or |dIb/dVbe|=Ib0/(Vcc-Vbe0) instead.
Therefore, if we use smaller Vcc, we get larger change in Ib, hence in Ic, hence in Vce, hence in bias point, just like MrAl has demonstrated it:
We can now immediately see that Vcc is in the denominator, so as Vcc gets larger the
magnitude of the change in Ib gets smaller. Thus a higher Vcc leads to less of a change
in Ib which leads to less of a change in Ic, which means less change in the bias point.
One thing that this circuit design is not a 'good' one is that the collector current depends heavily on the value of beta. Improvements can be made like audioguru suggested:
A properly biased transistor has a voltage divider feeding base current and an emitter resistor providing DC negative feedback.