Derivatives

Learning goals

• Define the derivative as a limit; interpret as slope and instantaneous rate.
• Differentiate powers, exponentials, logs, trig; use product, quotient, chain rules.
• Implicit differentiation; related rates as a later bridge to applications unit.
• Use linear approximation $L(x)=f(a)+f'(a)(x-a)$ near a base point $a$.

Concept sheet

Power rule: $\dfrac{d}{dx}\big(x^n\big) = n x^{n-1}$ for integer $n$ (extend to rationals as your course allows).

Exponential / log: $\dfrac{d}{dx} e^x = e^x$; $\dfrac{d}{dx} \ln x = \dfrac{1}{x}$ for $x \gt 0$.

Trig: $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$.

Combination rules ($u$ and $v$ are functions of $x$)

Same rules with $f,g$ notation: e.g. product is $(fg)'$ with $u=f$, $v=g$.

• Sum / difference: $(u \pm v)' = u' \pm v'$
• Constant multiple: $(c\,u)' = c\,u'$ for constant $c$
• Product rule: $(uv)' = u\,v' + v\,u'$  (same as $uv' + vu'$)
• Quotient rule: $\displaystyle \left(\frac{u}{v}\right)' = \frac{u'\,v - u\,v'}{v^{2}}$  (for $v \neq 0$)
• Chain rule: if $y = u(v(x))$, then $\displaystyle \frac{dy}{dx} = \frac{du}{dv}\cdot\frac{dv}{dx}$. With primes: $\big(u(v)\big)' = u'(v)\,v'$

Implicit (extra): differentiate both sides with respect to $x$, treating $y=y(x)$; collect $y'$.

Worked patterns

Pattern A — chain on chain (“peeling”)

$\dfrac{d}{dx} \sin(3x^2+1) = \cos(3x^2+1)\cdot 6x$.

Pattern B — product + chain

$\dfrac{d}{dx}\big(x^2 e^{-x}\big) = 2x\,e^{-x} + x^2(-e^{-x}) = e^{-x}(2x-x^2)$.

Pattern C — logarithmic differentiation (when handy)

For $y = x^x$ (Calc 1 extra): $\ln y = x\ln x$, differentiate: $\dfrac{y'}{y} = \ln x + 1$, so $y' = x^x(\ln x + 1)$.

Practice

Original drills — use Check when available, then Show solution. A quick celebration plays on correct auto-checks (respects reduced motion).

Homework / practice

Syllabus order §2.7–§3.8 (before related rates).

Section	Link	Focus
2.7 — Derivatives and rates of change	Open PDF	Limit definition of derivative, interpretations
2.8 — The derivative as a function	Open PDF	$f'$, differentiability vs continuity
Review 1 (pack)	Open PDF	Exam I: Appendix J.1 through §2.8 (per syllabus)
3.1 — Polynomials & exponentials	Open PDF	Power rule, $e^x$
3.2 — Product & quotient rules	Open PDF	Combinations of elementary functions
3.3 — Trigonometric derivatives	Open PDF	Sine, cosine, etc.
3.4 — Chain rule	Open PDF	Composition structure
3.5 — Implicit differentiation	Open PDF	$\dfrac{dy}{dx}$ from implicit $F(x,y)=0$
3.6 — Logarithmic differentiation	Open PDF	$\ln x$, $\log_a x$, logs of products
3.7 — Natural & social sciences	Open PDF	WebAssign export title truncated in filename
3.8 — Exponential growth & decay	Open PDF	WebAssign export title truncated in filename

Mistakes & checks

• Chain rule: multiply by inner derivative — easy to drop.
• Quotient rule sign: “low d-high minus high d-low” — verify with product rule rewrite.
• Differentiable at $a$ $\Rightarrow$ continuous at $a$; converse is false (e.g. $|x|$ at $0$).

Slide appendix (optional)

Optional condensed slide PDFs — see materials/README.md.

Quiz mode

Homework PDF quiz: each card links to the same WebAssign export filenames as the semester map (materials/). A session shows 20 shuffled cards from 45 PDF-linked slots; exact problem wording stays in the PDF unless you merge transcripts per quiz/transcripts/README.md. Regenerate: python3 scripts/gen_hw_quiz.py.