#### To determine

**To give:**

examples of derivative, interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences.

#### Explanation

**a) Physics:**

If s=f(t) is the position function of a particle that is moving in a straight line, then ΔsΔt represents the average velocity over a time period Δt, and v=dsdt represents the instantaneous velocity ( the rate of change of displacement with respect to time.).The instantaneous rate of change of velocity with respect to time is acceleration:

at=v't=s"(t).

**b) Chemistry:**

If a given substance is kept at a constant temperature, then its volume V depends on its pressure P.We can consider the rate of change of volume with respect to pressure- namely, the derivative dVdP.Since with increase in P, V decreases,

dVdP<0 To get the compressibility, we need to add a minus sign and divide the derivative by the volume V:

isothermal compressibility=Β=-1V.dVdP

Thus Β measures how fast, per unit volume, the volume of a substance decreases as the pressure on it increases at constant temperature.

For instance, if =5.3P, the volume V (in cubic meters) of a sample of air at 250C is related to the pressure P(in kilopascals). The rate of change of V, when P=50kPa is

dVdp|P=50=-5.3P2=-5.32500=-0.00212m3/kPa

The compressibility at that pressure is

Β=-1V.dVdP=0.002125.350=0.02(m3kPa)/m3

**c) Biology:**

Consider a vein of artery with blood flowing through it. Consider the blood vessel as a cylindrical tube with radius R and length l as illustrated in the below figure.

The velocity v of the blood is greatest along the central axis of the tube because of friction at the walls of the tube. The velocity decreases as the distance r from the axis increases, until v becomes 0 at the wall. The relationship between v and r is given by the law of laminar flow discovered by the French physician Jean-Louis-Marie Poiseuille in 1840. This low states that v=P4ηl(R2-r2) where η the viscosity of the blood and P is the pressure difference between the ends of the tube. If P and l are constant, then v is a function of r with domain [0,R].

The average rate of change of the velocity as we move from r=r1 outward to r=r2 is given by

ΔvΔr=vr2-vr1r2-r1 If we let Δr→0, we obtain the velocity gradient, that is, the instantaneous rate of change of velocity with respect to r.

velocity gradient=lim Δr→0ΔvΔr=dvdr

Using equation 1, we obtain

dvdr=P4ηl0-2r=-Pr2ηl Consider a smaller human artery. For this, we can take η=0.027,R=0.008 cm,l=2cm, and P=4000 dynes/cm3, which gives v=400040.0272(0.000064-r2)≈1.85×104(6.4×10-5-r2)

**d) Economics:**

Consider a company incurs the total cost as C(x) for producing x units of any commodity. The function C is called a **cost function.** If we increase the number of items produced from x1 to x2, then the additional cost is ΔC=Cx2-C(x1), and the average rate of change of the cost is

ΔCΔx=Cx2-Cx1x2-x1=Cx1+Δx-Cx1Δx

The rate of change of cost to the number of items produced, is called the **marginal cost **by economists. The limit of this quantity as Δx→0, is

marginal cost=lim Δx→0ΔCΔx=dCdx

Taking Δx=1 and n large ( so than Δx is small compared to n), we have

C'n≈Cn+1-C(n)

The cost of producing one more unit [ the n+1st unit], is almost equal to the marginal cost of producing n units.

It is often appropriate to represent a total cost function by a polynomial

Cx=a+bx+cx2+dx3

**e) Other sciences:**

In psychology, the learning theory study or the so-called learning curve, graphs the performance P(t) of someone’s learning a skill as a function of the training time t. Of particular interest is the rate at which performance improves as time passes, that is,dP/dt.