Chapter 6 — Key Takeaways
A structured recap of The Derivative: Definition, Notation, and What It Means. Use it to review before the quiz and before moving on to the differentiation rules of Chapter 7.
1. The derivative is a function, not a number
The decisive move of this chapter (§6.2): let the point of evaluation become a variable. The pointwise slope $f'(a)$ becomes a whole new function
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h},$$
defined on the set $D$ where the limit exists — the domain of differentiability. This domain can be smaller than the domain of $f$ itself: $f(x) = \sqrt{x}$ is defined at $0$, but $f'$ is not.
2. Computing $f'$ from the definition: the four-step ritual (§6.2)
- Write the difference quotient $\dfrac{f(x+h) - f(x)}{h}$.
- Expand/simplify the numerator until a factor of $h$ appears.
- Cancel the $h$ to clear the $0/0$ form.
- Take the limit as $h \to 0$.
The starter catalog you should be able to reproduce by hand:
| $f(x)$ | $f'(x)$ | trick |
|---|---|---|
| $c$ | $0$ | numerator is $0$ |
| $x$ | $1$ | numerator is $h$ |
| $x^2$ | $2x$ | expand, cancel $h$ |
| $x^3$ | $3x^2$ | expand, cancel $h$ |
| $1/x$ | $-1/x^2$ | common denominator |
| $\sqrt{x}$ | $1/(2\sqrt{x})$ | rationalize with the conjugate |
| $e^x$ | $e^x$ | use $\lim_{h\to0}\frac{e^h-1}{h}=1$ |
3. Graphing $f'$ straight from $f$ (§6.4)
The translation dictionary:
- $f$ rising $\Rightarrow$ $f' > 0$; steeper rise $\Rightarrow$ larger $f'$.
- $f$ falling $\Rightarrow$ $f' < 0$.
- $f$ has a horizontal tangent $\Rightarrow$ $f' = 0$.
Points where $f' = 0$ or $f'$ fails to exist are the critical points of $f$ — the candidates for local maxima and minima, and the engine of the optimization to come in Chapter 10.
4. Differentiability is strictly stronger than continuity (§6.3)
Theorem: if $f'(a)$ exists, then $f$ is continuous at $a$.
The arrow runs one way only. Every break in a graph kills the derivative for free, but continuity alone never guarantees a derivative — $|x|$ is continuous at $0$ yet has no derivative there.
| Continuous? | Differentiable? | Example |
|---|---|---|
| No | No (automatically) | step function at its jump |
| Yes | No | $\lvert x\rvert$ at $0$ |
| Yes | Yes | polynomials, $\sin x$, $e^x$ |
5. The four ways differentiability fails (§6.7)
- Corner — one-sided slopes exist but disagree ($|x|$ at $0$).
- Cusp — one-sided slopes run to $+\infty$ and $-\infty$ ($x^{2/3}$ at $0$).
- Vertical tangent — slope $\to \pm\infty$ from both sides ($x^{1/3}$ at $0$; also $\sqrt{x}$ at $0$).
- Discontinuity — no continuity, so no derivative (by §6.3).
A function differentiable at every point of an interval is smooth there. The Weierstrass function (§6.9) is continuous everywhere yet differentiable nowhere — proof that the smooth functions of a first course are the well-behaved minority.
6. Higher-order derivatives (§6.6)
Differentiate $f'$ to get $f''$, again for $f'''$, and so on. Two readings of the second derivative:
- Kinematics: if $s(t)$ is position, then $s' = $ velocity, $s'' = $ acceleration, $s''' = $ jerk, $s^{(4)} = $ snap. Newton's law $F = m\,s''$ lives in the second row.
- Geometry: $f'' > 0 \Rightarrow$ concave up; $f'' < 0 \Rightarrow$ concave down; a sign change of $f''$ marks an inflection point.
A degree-$n$ polynomial has $f^{(n+1)}(x) = 0$ identically — each differentiation lowers the degree by one. (E.g., $f = x^4 \Rightarrow f^{(4)} = 24 = 4!$, $f^{(5)} = 0$.)
7. Three notations, all in use (§6.9)
| Tradition | First derivative | Higher | Best for |
|---|---|---|---|
| Lagrange | $f'(x)$ | $f''(x), f^{(n)}(x)$ | stating theorems |
| Leibniz | $\dfrac{df}{dx}$ | $\dfrac{d^n f}{dx^n}$ | chain rule, integration |
| Newton | $\dot{x}$ | $\ddot{x}, \dddot{x}$ | time derivatives in physics |
The operator $Df = \frac{d}{dx}f$ treats differentiation as a machine; it is linear, $D(af + bg) = a\,Df + b\,Dg$, which underlies nearly every rule of Chapter 7.
8. Gradient descent — the chapter's anchor example (§6.8)
To minimize $f(x)$ without solving $f'(x) = 0$ algebraically, walk downhill. The derivative is a compass pointing uphill, so step the opposite way:
$$x_{n+1} = x_n - \alpha\, f'(x_n).$$
The minus sign is the whole trick, and it is self-correcting: $f' > 0$ pushes you left, $f' < 0$ pushes you right, both downhill. The learning rate $\alpha$ must be chosen with care — too large overshoots and diverges, too small crawls. This single idea, generalized to the multivariable gradient $\nabla f$ in Chapter 30, is the algorithm that trains essentially every modern neural network.
Common Errors to Avoid
- Plugging in $h = 0$ before cancelling the $h$ — $0/0$ is a signal to do more algebra, not an answer (§6.2).
- Assuming "continuous at $a$" implies "$f'(a)$ exists" — corners, cusps, and vertical tangents disprove it (§6.7).
- Treating $\frac{dy}{dx}$ as a literal fraction you may cancel — in single-variable calculus it is one indivisible symbol (§6.9).
- Confusing the domain of $f$ with the (possibly smaller) domain of $f'$ (§6.2).
Skills You Should Now Have
- Compute $f'(x)$ from the limit definition for any elementary function.
- Sketch $f'$ given only the graph of $f$, and locate critical points.
- Identify and classify every point where a function is non-differentiable.
- Compute and interpret higher-order derivatives (acceleration, jerk, concavity).
- Move fluently among Lagrange, Leibniz, Newton, and operator notation.
- Run gradient descent by hand and in Python, and explain the role of $\alpha$.
Connections Across the Book
The derivative now branches into nearly everything ahead: the differentiation rules of Chapter 7, curve sketching in Chapter 9, optimization in Chapter 10, linear approximation and Newton's method in Chapter 11, antiderivatives in Chapter 12, differential equations in Chapter 19, the multivariable gradient and gradient descent's full form in Chapter 30, and the vector calculus of Part VII. The derivative measures change — and calculus is the mathematics of change.
What's Next
Chapter 7 replaces the limit-definition ritual with a toolkit of rules — power, sum, product, quotient, and chain — that let you differentiate any elementary function in seconds, often in your head. Everything you computed laboriously here becomes a one-line move.