Derivative Study Guide

📖 Core Concepts Derivative – instantaneous rate of change; slope of the tangent line; best linear approximation near a point. Limit definition – \(f'(a)=\displaystyle\lim{h\to0}\frac{f(a+h)-f(a)}{h}\) (exists ⇔ function is differentiable at \(a\)). Continuity ⇒ Differentiability – Differentiability at \(a\) guarantees continuity at \(a\), but not vice‑versa. Higher‑order derivatives – Re‑apply the derivative operator: \(f',\;f'',\;f^{(n)}\). Antiderivative – Function \(F\) with \(F'=f\); families differ by a constant \(C\). Fundamental Theorem of Calculus (FTC‑I) – If \(F\) is an antiderivative of \(f\) on \([a,b]\), then \(\displaystyle\inta^b f(x)\,dx = F(b)-F(a)\). Multivariable derivatives – Gradient \(\nabla f\), directional derivative \(D{\mathbf u}f\), Jacobian matrix \(J{\mathbf F}\), total derivative (linear map approximating \(f\) in all directions). --- 📌 Must Remember Constant rule: \(\displaystyle\frac{d}{dx}(c)=0\). Power rule: \(\displaystyle\frac{d}{dx}x^{n}=n x^{\,n-1}\) (any real \(n\)). Exponential rule: \(\displaystyle\frac{d}{dx}e^{x}=e^{x}\). Log rule: \(\displaystyle\frac{d}{dx}\ln x=\frac{1}{x}\). Trig rules: \(\frac{d}{dx}\sin x=\cos x,\;\frac{d}{dx}\cos x=-\sin x,\;\frac{d}{dx}\tan x=\sec^{2}x\). Sum/Difference: \((u\pm v)'=u'\pm v'\). Product rule: \((uv)'=u'v+uv'\). Quotient rule: \(\displaystyle\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^{2}},\;v\neq0\). Chain rule: If \(y=f(g(x))\), then \(y' = f'(g(x))\cdot g'(x)\). Differentiability class: \(f\in C^{n}\) ⇔ the first \(n\) derivatives exist and are continuous. Smoothness: \(f\) is smooth ⇔ all orders of derivatives exist (belongs to \(C^{\infty}\)). --- 🔄 Key Processes Compute a derivative Identify the rule(s) needed (power, product, chain, etc.). Apply rules step‑by‑step, simplifying after each application. Check differentiability Verify continuity at the point. Look for kinks, cusps, jumps, or vertical tangents. Find an antiderivative Recognize the basic form (power, exponential, trig, log). Add constant \(C\). Evaluate a definite integral (FTC‑I) Find any antiderivative \(F\) of \(f\). Compute \(F(b)-F(a)\). Compute a gradient Take partial derivative w.r.t. each variable: \(\nabla f = \big(\partial f/\partial x{1},\dots,\partial f/\partial x{n}\big)\). Directional derivative Normalize direction vector \(\mathbf u\). Compute \(D{\mathbf u}f = \nabla f \cdot \mathbf u\). --- 🔍 Key Comparisons Power rule vs. Exponential rule Power: derivative lowers exponent by 1 (\(x^{n}\to n x^{n-1}\)). Exponential: derivative leaves the function unchanged (\(e^{x}\to e^{x}\)). Partial vs. Total derivative Partial \(\partial f/\partial x{i}\): varies one variable, holds others fixed. Total derivative (Jacobian): captures how all variables change simultaneously; exists only if all partials are nicely behaved. Leibniz vs. Prime notation Leibniz: \(\frac{dy}{dx}\) emphasizes “change of \(y\) with respect to \(x\)”. Prime: \(y'\) is compact, used when the independent variable is clear. --- ⚠️ Common Misunderstandings “All continuous functions are differentiable.” Counter‑example: \(f(x)=|x|\) is continuous at 0 but not differentiable there. “If the limit in the definition exists, the derivative must be finite.” Vertical tangent yields an infinite slope → derivative does not exist as a finite number. “Chain rule only applies to compositions of different functions.” It works for any composition, even when the inner function is a simple power or constant multiple. “The derivative of a constant is the constant itself.” It is zero; the constant’s slope is flat. --- 🧠 Mental Models / Intuition Tangent line = best linear sketch – near any point, the curve looks like its tangent; use this to approximate small changes: \(\Delta y \approx f'(a)\Delta x\). Derivative as a “speedometer” – first derivative tells how fast a quantity changes; second derivative tells how that speed itself changes (acceleration). Product rule as “product of movers” – when two things move, total change = change of first × second + first × change of second. Chain rule as “gear train” – inner function turns the “gear”, outer function turns the “output”; multiply the two rates. --- 🚩 Exceptions & Edge Cases Vertical tangent – slope \(\to\pm\infty\); derivative undefined (e.g., \(f(x)=\sqrt[3]{x}\) at \(x=0\)). Cusp/kink – left‑hand and right‑hand slopes differ (e.g., \(f(x)=|x|\) at 0). Jump discontinuity – function jumps; no tangent line → not differentiable. Higher‑order derivative may fail – a function can be differentiable once but not twice (e.g., \(f(x)=|x|^{3/2}\) has a derivative at 0 but its second derivative does not exist). --- 📍 When to Use Which Power vs. Exponential rule – Use Power when the base is a variable raised to a constant exponent; use Exponential when the base is the constant \(e\). Product vs. Quotient rule – Use Product for \(u\cdot v\); use Quotient for \(\frac{u}{v}\) (remember \(v\neq0\)). Chain rule – Whenever you see a composition \(f(g(x))\) (nested functions, trig of a polynomial, etc.). Gradient vs. Directional derivative – Compute full gradient first; then dot with any unit direction \(\mathbf u\) to get the directional derivative. FTC‑I – Use when the problem asks for area/definite integral and an antiderivative is easy to find. --- 👀 Patterns to Recognize “\(f'(x)=0\) at extrema” – Look for zero derivatives to spot local maxima/minima (provided derivative changes sign). Repeated factors → product rule – When a function is a product of several simple pieces, apply product rule iteratively. Nested functions → chain rule – Any expression like \(\sin(x^{2})\), \(\ln(1+x^{3})\), etc., signals chain rule. Symmetry in multivariable problems – If a function is symmetric in variables, gradient components often share a pattern. --- 🗂️ Exam Traps Confusing \(f'(x)\) with \(f(x)\) – Remember the derivative is a different function; don’t substitute the original expression when the problem asks for the slope. Dropping the denominator in the quotient rule – Students often write \((u'v-uv')\) without dividing by \(v^{2}\). Mis‑applying the chain rule to sums – Only compositions need the chain rule; a simple sum uses the sum rule. Assuming continuity ⇒ differentiability – Watch for absolute‑value or piecewise definitions at the point of interest. For directional derivatives, forgetting to normalize \(\mathbf u\) – The direction vector must be a unit vector; otherwise the result is scaled incorrectly. Neglecting the constant \(C\) in antiderivatives – When asked for “the antiderivative”, include “\(+C\)” to capture the full family.