   Chapter 3.2, Problem 59E

Chapter
Section
Textbook Problem

The Michaelis-Menten equation for the enzyme chymotrypsin is v = 0.14 [ S ] 0.015 + [ S ] where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv/d[S] and interpret it.

To determine

To find: The value of dvd[S].

Explanation

Given:

The equation for the enzyme chymotrypsin is v=0.14[S]0.015+[S]

Derivative rule:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Constant multiple rule: ddx(cf)=cddx(f)

Calculation:

The derivative of the function v=0.14[S]0.015+[S] is dvd[S] which is obtained as follows,

dvd[S]=dd[S](0.14[S]0.015+[S])

Apply the Quotient rule (1) and simplify the terms,

dvd[S]=(0.015+[S])dd[S](0.14[S])(0.14[S])dd[S](0.015+[S])[0.015+[S]]2

Apply the sum rule (3) and constant multiple rule (4),

dvd[S]=[(0

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