5.4: The derivative of ekt.
( \newcommand{\kernel}{\mathrm{null}\,}\)
We have found the derivative of et. Often, however, the function of interest is of the form C ekt where C and k are constants. In Example 5.3.1 of bacterial growth,
Abs =0.0174e0.02176 Time
the constant C=0.0174 and the function kt=0.02176 Time. We develop a formula for [ekt]′.
[ekt]′=[(et)k]′(i)=k(et)k−1[et]′(ii)=k(et)k−1et(iii)=kekt
Because Equation ??? [ekt]′=kekt, is used so often, we call it another Primary Formula even though we developed it without direct reference to the Definition of Derivative. Should you be limited to a single derivative rule, in the life sciences choose the ekt Rule – exponential functions are ubiquitous in biology.
E(t)=ekt⇒E′(t)=ektk[ekt]′=kekt
Explore 5.4.1 Were we to derive [ekt]′=kekt
[ekt]′=limb→aekb−ekab−a(i)=limb→aekb−ekakb−kak(ii)=ekak(iii)
The assertion in step (iii) that
limb→aekb−ekakb−ka=eka
is correct, puzzles some students, and is worth your thought.
We can now differentiate functions like P(t)=5t7+3e2t.
P′(t)=[5t7+3e2t]′A symbolic identity.=[5t7]′+[3e2t]′Sum Rule=5[t7]′+3[e2t]′Constant Factor Rule=5×7t6+3[e2t]′Power Rule=35t6+3e2t2ekt Rule=35t6+6e2t
Example 5.4.1 We can also compute E′(t) for E(t)=2t.
[2t]′=[(eln2)t]′=[e(ln2)t]′=e(ln2)tln2=2tln2
We have an exact solution for the first problem of this Chapter, which was to find E′(2) for E(t)=2t. The answer is 22ln2=4ln2. Also, E′(0)=20ln2=ln2 which answers another question from early in the chapter.
More generally, for b>0,
[bt]′=[(elnb)t]′(i)=[e(lnb)t]′(ii)=e(lnb)tlnb(iii)=btlnb(iv)
We summarize this information:
[bt]′=btlnb for b>0
Explore 5.4.2 This is very important. Show that if C and k are constants and P(t)=C ekt then P′(t)=k P(t).
Exercises for Section 5.4, The derivative of ekt.
Exercise 5.4.1 Give reasons for the steps (i) − (iii) in Equation ??? showing that [ekt]′=ektk.
Exercise 5.4.2 Give reasons for the steps (i) − (iv) in Equation ??? showing that [bt]′=btlnb.
Exercise 5.4.3 The function bt for b=1 is a special exponential function. Confirm that the derivative equation [bt]′=btlnb is valid for b=1. Draw some graphs of bt for b=1 and its derivative.
Exercise 5.4.4 Use one rule for each step and identify the rule to differentiate
- P(t)=3e5t+π
- P(t)=e22+t33
- P(t)=5t
- P(t)=e2te3t
Simplify Part d before differentiating.
Exercise 5.4.5 Compute y′(x) or assert that you do not yet have forumlas to compute y′(x) for
- y(x)=e5x
- y(x)=e−3x
- y(x)=e√x
- y(x)=(ex)2
- y(x)=(e√x)2
- y(x)=(e−x)2
- y(x)=ex+e−x2
- y(x)=ex−e−x2
- y(x)=5e−0.06x+3e−0.1x
- y(x)=e(x2)
- y(x)=√ex
- y(x)=8e−0.0001x−16e−0.001x
- y(x)=e5
- y(x)=√e
- y(x)=10x
- y(x)=10−x
- y(x)=x2+2x
- y(x)=(e5x+e−3x)5
Exercise 5.4.6 Interpret et2 as e(t2) . Argue that
limb→ae(b2)−e(a2)b2−a2=e(a2)
What is the ambiguity in the notation ea2. (Consider 432.) Use parenthesis, they are cheap. However, common practice is to interpret (e ^{t ^{2}}\) as e(t2).
Exercise 5.4.7 Argue that
limb→ae√b−e√a√b−√a=e√a
Exercise 5.4.8 Review the method in Explore 5.4.1 and the results in Exercises 5.4.6 and 5.4.7. Use Definition 3.22,
F′(a)=limb→aF(b)−F(a)b−a, to compute E′(a) for
- E(t)=e2t
- E(t)=e2√t
- E(t)=e−t
- E(t)=e2
- E(t)=e1t
- E(t)=e−t2
Exercise 5.4.9 Consider the kinetics of penicillin that is taken as a pill in the stomach. The diagram in Figure Ex. 5.4.9(a) may help visualize the kinetics. We will find in Chapter 17 that a model of plasma concentration of antibiotic t hours after ingestion of an antibiotic pill yields an equation similar to
C(t)=5e−2t−5e−3tμg/ml
A graph of C is shown in Figure Ex. 5.4.9. At what time will the concentration reach a maximum level, and what is the maximum concentration achieved?
As we saw in Section 3.5.2 and may be apparent from the graph in Figure Ex. 5.4.9, the highest concentration is associated with the point of the graph of C at which C′=0; the tangent at the high point is horizontal. The question, then, is at what time t is C′(t)=0 and what is C(t) at that time?
Figure for Exercise 5.4.9 (a) Diagram of compartments for oral ingestion of penicillin. (b) Graph of C(t)=5e−2t−5e−3t representative of plasma penicillin concentration t minutes after ingestion of the pill.
Exercise 5.4.10 Plasma penicillin concentration is
P(t)=5e−0.3t−5e−0.4t
t hours after ingestion of a penicillin pill into the stomach. A small amount of the drug diffuses into tissue and the tissue concentration, C(t), is
C(t)=−e−0.3t+0.5e−0.4t+0.5e−0.2tμg/ml
- Use your technology (calculator or computer) to find the time at which the concentration of the drug in tissue is maximum and the value of C at that time.
- Compute C′(t) and solve for t in C′(t)=0. This is really bad, for you must solve for t in 0.3e−0.3t−0.2e−0.4t−0.1e−0.2t=0 Try this: Let Z=e−0.1t then solve 0.3Z3−0.2Z4−0.1Z2=0.
- Solve for the possible values of Z. Remember that Z=e−0.1t and solve for t if possible using the possible values of Z.
- Which value of t solves our problem?