Calculus I · The Function That Is Its Own Slope

Where this is heading

Two threads to carry forward.

The fact that $e^x$ is its own derivative is not just convenient — it is about to make $e^x$ the single cleanest example in the next chapter. When we learn to rebuild any function out of its own derivatives (Taylor’s idea, Chapter 6), $e^x$ produces the most beautiful pattern of all, precisely because differentiating it changes nothing.

And the same self-similarity is why $e^x$ shows up everywhere once you reach machine learning: it is inside the “softmax” that turns scores into probabilities, and inside the bell curve that describes noise and uncertainty. You are not learning $e^x$ because it is on a test. You are learning the one function that nature and computation both reach for first.

Problems

Six tiers. Fluency and Practice are doing* — drill the derivatives and log rules until automatic, then apply them. Concept & counterexample and ◆ Diamond are thinking — traps, constructions, and the proofs (including the honest treatment of where $e$ comes from). Teach it is teaching. Looking ahead reaches into Chapter 6. Do all of Fluency; don’t skip Concept or Teach it.*

Fluency — do these fast, until they’re automatic

1. Differentiate: (a) $e^{5x}$; (b) $e^{-x}$; (c) $3e^{2x}$; (d) $e^{x}+x^3$; (e) $e^{x/2}$; (f) $7e^{-0.1x}$.
2. Differentiate: (a) $\ln x$; (b) $\ln(3x)$; (c) $5\ln x$; (d) $\ln(x)+x^2$.
3. Use the log rules to rewrite as sums/differences/multiples: (a) $\ln(6)$; (b) $\ln\!\frac{x}{5}$; (c) $\ln(x^3)$; (d) $\ln(2x^2)$.
4. Evaluate without a calculator: (a) $\ln 1$; (b) $\ln e$; (c) $\ln e^4$; (d) $e^{\ln 7}$; (e) $\ln e^{-2}$.
5. A colony follows $N(t)=200\,e^{0.4t}$ ($t$ in hours). Find $N(0)$ and $N(5)$.
6. A drug has half-life $4$ hours; you start with $80$ mg. How much remains after $4$, $8$, and $12$ hours? (In your head.)

Practice — choose the rule and apply it

1. Differentiate, naming outer and inner functions first: (a) $e^{3x^2}$; (b) $\ln(x^2+1)$; (c) $e^{-x^2}$; (d) $\ln(\sqrt{x})$.
2. Show by differentiating that $N(t)=N_0e^{kt}$ satisfies $\tfrac{dN}{dt}=kN$, and state in one sentence what $N_0$ and $k$ mean.
3. A radioactive isotope decays with $\lambda=0.05$ per year. Find its half-life, then the fraction remaining after $30$ years.
4. $1000 grows at a continuously compounded $5\%$, so $A(t)=1000\,e^{0.05t}$. How long until it doubles? (Then explain why $\ln 2\approx0.69$ makes “$70\div\text{rate}\%$” a good mental shortcut.)
5. Two drugs: A has half-life $2$ h, B has half-life $6$ h. Write each as $N_0e^{-\lambda t}$ (find each $\lambda$), and find how long until each falls to $10\%$ of its dose.
6. A pH is defined by $\text{pH}=-\log_{10}[\text{H}^+]$. If one solution has $[\text{H}^+]$ one hundred times another’s, how far apart are their pH values? (One line — this is the log-scale point.)
7. Differentiate $h(x)=e^{2x}+\ln(5x)$, then find the slope of $h$ at $x=1$.

Concept & counterexample — think; state true/false and justify, or build the example

1. 1. $\tfrac{d}{dx}e^{x}=xe^{x-1}$ (treating $e^x$ “like a power”). (b) $\ln(A+B)=\ln A+\ln B$. (c) The graph of $e^x$ has the same slope as height at every point. (d) $2^x$ grows but its slope-at-zero is less than $1$.
2. Explain in one or two sentences why $e$, not $10$, is called the natural base — what specifically is natural about it?
3. Construct / identify: among $f(x)=2^x$, $e^x$, $3^x$, which one has the property that its derivative equals itself, and which two do not? For the two that don’t, what does each derivative equal?
4. Find the error. A student writes: “$\tfrac{d}{dx}\ln(x^2) = \tfrac{1}{x^2}$, by the rule $\tfrac{d}{dx}\ln x=\tfrac1x$.” Identify the missing step and give the correct derivative two different ways (chain rule, and first using a log rule).
5. Is it possible? A quantity decaying by first-order kinetics that reaches exactly zero in finite time. Argue from the form $N_0e^{-\lambda t}$ why or why not. (What does this say about “when is the drug completely gone?”)
6. The chapter took out three “loans” about $e$. State them in your own words, and say which chapter repays each.

◆ Diamond — Layer 2: prove it

1. ◆ Prove $\tfrac{d}{dx}\ln x=\tfrac1x$ using only that $\ln$ is the inverse of $e^x$ and the chain rule (reproduce the text’s three-line argument from scratch).
2. ◆ Prove $M(a)=\ln a$ by writing $a^x=e^{(\ln a)x}$ and differentiating. Then deduce $\tfrac{d}{dx}a^x=(\ln a)a^x$ and check it gives the right answer at $a=e$.
3. ◆ Derive the half-life formula $t_{1/2}=\dfrac{\ln 2}{\lambda}$ from $N_0e^{-\lambda t}$, justifying each logarithm step. Then prove the “memoryless” property: the time to fall from any level to half of it is always $t_{1/2}$, independent of where you start.
4. ◆ Using the slope-at-the-axis idea, explain why $M(4)>1$ and $M(2)<1$, and why this forces a base with $M(a)=1$ to exist between them. Name the property of $M$ you’re assuming (it’s one of the chapter’s loans).
5. ◆ Prove $\tfrac{d}{dx}\ln(kx)=\tfrac1x$ for any constant $k>0$ — and explain in one sentence why the constant vanishes (what does multiplying the input by $k$ do to the graph of $\ln$?).
6. ◆◆ Uniqueness. Suppose $f'(x)=k\,f(x)$ for all $x$, with $f(0)=N_0$. Show $g(x)=f(x)e^{-kx}$ has $g'(x)=0$ everywhere, conclude $g$ is constant, and deduce $f(x)=N_0e^{kx}$. (This proves the exponential is the solution to the growth/decay equation, not merely a solution.)
7. ◆◆ Logarithmic differentiation preview: by writing $a^x=e^{(\ln a)x}$, and more generally any $u(x)^{v(x)}=e^{v(x)\ln u(x)}$, differentiate $x^x$ for $x>0$. (You’ll get a clean answer; notice the trick is to never differentiate a “variable-to-a-variable” directly — always route through $e$ and $\ln$ first.)

Teach it — explain it, write it up, or fix it for someone else

1. In three or four sentences to a friend, explain why $e^x$ is “the function that is its own slope,” starting from the question “what function is proportional to its own rate of change?” — not from the formula.
2. Write a worked solution to “a drug with half-life $3$ hours: how long until $5\%$ remains?” narrating each step (set up the model, find $\lambda$, take the log) so a peer could redo it for any half-life.
3. Debug your classmate. They claim “$\tfrac{d}{dx}e^{2x}=e^{2x}$, because $e$ to anything is its own derivative.” Pinpoint the error, name the rule they skipped, and give the correct derivative.
4. Design a problem. Invent a continuously-compounded investment problem whose answer is “it doubles in exactly $10$ years,” stating the rate you need and writing the solution key.
5. In two or three sentences, teach why mathematicians prefer $\ln$ (base $e$) over $\log_{10}$, even though $\log_{10}$ seems friendlier — what does base $e$ buy you that base $10$ doesn’t? (Hint: look at the derivatives.)

Looking ahead — Chapter 6 in your hands early

1. $e^x$ is its own derivative, so its first, second, third… derivatives are all $e^x$, which all equal $1$ at $x=0$. In Chapter 6 you’ll build a function out of its derivatives at a point. Guess the first four terms of the polynomial that best matches $e^x$ near $0$. (Try matching value, slope, and curvature at $x=0$ — you may recognize the pattern $1+x+\tfrac{x^2}{2}+\cdots$.)
2. Compute $e^{0.1}$ on a calculator, then compare it to $1+0.1$ and to $1+0.1+\tfrac{0.1^2}{2}$. Which is closer? (You’re watching a Taylor approximation get better — Chapter 6.)
3. The clean, no-loans construction of $e$ defines $\ln x=\displaystyle\int_1^x \tfrac{1}{t}\,dt$ — an area. You don’t have integrals yet. Predict: if $\ln x$ is “the area under $\tfrac1t$ from $1$ to $x$,” what should $\ln 1$ be, and why should $\ln x$ increase as $x$ grows? (Chapter 10 makes this airtight and repays the chapter’s loans.)

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