Chapter 7 — Key Takeaways
This chapter turned differentiation from a limit you grind through (Chapter 6) into an algebra you apply by pattern recognition. Here is the complete recap, organized so you can study from it directly.
1. The Big Idea
The derivative is a machine that respects structure. Functions are built by adding, subtracting, multiplying, dividing, and composing simpler functions, and each of those operations has its own differentiation rule — derived once from the limit definition, then reused forever. Learn the rules, learn which one matches which structure, and you can differentiate any elementary function mechanically. The limit does its work inside each proof; afterward you never return to it.
2. The Combination Rules
| Rule | Statement | When to use |
|---|---|---|
| Power | $\dfrac{d}{dx}(x^n) = n\,x^{n-1}$ (every real $n$) | A variable raised to a constant |
| Constant multiple | $(cf)' = c\,f'$ | A function scaled by a constant |
| Sum / difference | $(f \pm g)' = f' \pm g'$ | Terms added or subtracted |
| Product | $(fg)' = f'g + fg'$ | Two factors multiplied |
| Quotient | $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$ | One function divided by another |
| Reciprocal | $\left(\dfrac{1}{g}\right)' = -\dfrac{g'}{g^2}$ | Special case of the quotient rule |
| Chain | $\big(f(g(x))\big)' = f'(g(x))\cdot g'(x)$ | A function inside a function |
Derivations to remember (not just the formulas):
- Power rule (Section 7.2) follows from the binomial expansion of $(x+h)^n$: every term except the first keeps a factor of $h$ and vanishes in the limit. It holds for every real exponent — integers, rationals, irrationals.
- Linearity (Section 7.3) — sums and constant multiples — is just the limit laws of Chapter 3 applied to the difference quotient. The derivative is a linear operator.
- Product rule (Section 7.4) comes from the add-and-subtract trick on $f(x+h)g(x)$; geometrically, a rectangle of sides $f$ and $g$ grows by $f\,dg + g\,df$, and the corner $df\,dg$ is second-order and vanishes.
- Quotient rule (Section 7.5) is the product rule in disguise: write $f/g = f\cdot g^{-1}$. It is not a new axiom.
- Chain rule (Section 7.6) says rates of change multiply through a chain of dependencies: if $y$ changes 3× as fast as $u$, and $u$ changes 5× as fast as $x$, then $y$ changes 15× as fast as $x$.
3. The Library of Standard Derivatives
Memorize these — they are the vocabulary that the combination rules operate on.
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| $c$ | $0$ | $\sin x$ | $\cos x$ |
| $x^n$ | $n\,x^{n-1}$ | $\cos x$ | $-\sin x$ |
| $e^x$ | $e^x$ | $\tan x$ | $\sec^2 x$ |
| $b^x$ | $b^x \ln b$ | $\sec x$ | $\sec x \tan x$ |
| $\ln x$ | $1/x$ $(x>0)$ | $\csc x$ | $-\csc x \cot x$ |
| $\log_b x$ | $1/(x\ln b)$ | $\cot x$ | $-\csc^2 x$ |
| $\arcsin x$ | $1/\sqrt{1-x^2}$ | $\arctan x$ | $1/(1+x^2)$ |
| $\sinh x$ | $\cosh x$ | $\cosh x$ | $\sinh x$ |
Worth deriving, not just memorizing (Section 7.7):
- $\frac{d}{dx}(\sin x) = \cos x$ falls out of the angle-sum identity plus the two foundational trig limits $\frac{\sin h}{h}\to 1$ and $\frac{\cos h - 1}{h}\to 0$ from Chapter 3. These hold only in radians.
- $\frac{d}{dx}(e^x) = e^x$ is the defining property of $e$: among all curves $b^x$, only $e^x$ has slope equal to its own height. The general rule $\frac{d}{dx}(b^x) = b^x\ln b$ follows by writing $b^x = e^{x\ln b}$ and applying the chain rule.
4. Logarithmic Differentiation (Section 7.9)
When the variable appears in both a base and an exponent (like $x^x$), neither the power rule nor the exponential rule applies. Take $\ln$ of both sides first, which turns exponents into coefficients and products into sums, then differentiate:
$$y = x^x \;\Rightarrow\; \ln y = x\ln x \;\Rightarrow\; \frac{y'}{y} = \ln x + 1 \;\Rightarrow\; y' = x^x(\ln x + 1).$$
The step $\frac{d}{dx}\ln y = \frac{y'}{y}$ is implicit differentiation (formalized in Chapter 8). Logarithmic differentiation also tames messy products and quotients, since $\ln\frac{uv}{w} = \ln u + \ln v - \ln w$.
5. Combining Rules — The Strategy (Section 7.8)
- Look at the whole function. Identify the outermost operation (sum, product, quotient, or composition).
- Apply the matching rule for that operation.
- The rule hands you smaller derivative subproblems — recurse on each.
- Stop when you reach a library function and write down its known derivative.
Example: $y = x^2\sin(3x+1)$ is outermost a product, so $y' = 2x\sin(3x+1) + x^2\cdot 3\cos(3x+1)$ — the second factor needs the chain rule inside the product rule.
6. Common Errors to Avoid
- Power rule on an exponential. $\frac{d}{dx}(2^x) \ne x\,2^{x-1}$. The variable is in the exponent, so $\frac{d}{dx}(2^x) = 2^x\ln 2$. Ask: is the variable in the base or the exponent?
- Product of derivatives. $(fg)' \ne f'g'$. Test on $f=g=x$: the false rule gives $1$, but $(x^2)' = 2x$. Keep both terms.
- Dropping the chain factor. $\frac{d}{dx}\sin(x^2) = 2x\cos(x^2)$, not $\cos(x^2)$ and not $\cos(2x)$. Always ask "times the derivative of the inside?"
- Misordering the quotient rule. $f'g - fg'$, not $fg' - f'g$ — the sign flips if you swap them. And square the denominator.
- Degrees vs. radians. Trig derivatives assume radians; degrees inject an unwanted $\pi/180$.
- Forgetting differentiability. The rules say nothing at corners ($|x|$ at $0$), cusps ($x^{2/3}$ at $0$), or discontinuities. Make sure the derivative exists before applying a rule.
7. Skills You Should Now Have
- Differentiate any elementary function — powers, roots, trig, exponentials, logs, and their combinations — mechanically.
- Recognize composite structure and apply the chain rule, including nested compositions.
- Combine multiple rules in a single problem by identifying the outermost operation first.
- Use logarithmic differentiation when the standard rules stall.
- Verify any hand computation with
sympy(Section 7.10), treating disagreements as bugs to hunt down — usually a dropped chain factor or a quotient-rule sign error.
8. Connections
- Backward: the rules are derived from the limit definition (Chapter 6) and the limit laws (Chapter 3).
- Forward: the chain rule powers implicit differentiation and related rates (Chapter 8), separable differential equations (Chapter 19), the multivariable gradient and backpropagation (Chapter 30), and the integral theorems of vector calculus (Part VII). From here on, differentiation is assumed — it is the tool, not the topic.
- Anchor examples: the chain rule is the engine behind backpropagation (Case Study 1, climaxing in Chapter 30), and the self-replicating derivative of $e^x$ underlies the decay and dosing models of Case Study 2 (developed fully in Chapter 19).
What's Next
Chapter 8 develops two essential applications of the chain rule: implicit differentiation (for curves like $x^2 + y^2 = 1$ where $y$ is not isolated) and related rates (where two quantities change together in time). By the end of Part II (Chapter 12), you will have every differentiation tool the rest of the book relies on.
A reflection. The chain rule is the central rule of differential calculus — every other rule is convenience; the chain rule is necessity. Differentiation should feel as automatic as multiplication by the time you finish this chapter. If it still feels effortful, do more exercises. That fluency unlocks everything that comes after.