Lesson 6-1: Derivatives – Concept & Geometric Meaning

Appendix Set A: Definition Mastery

1. Evaluate a limit-difference quotient for a quadratic at a point
2. Explain when the limit fails to exist (corner/cusp/vertical tangent)
3. Relate average vs instantaneous rate with a small time window
4. Units sanity-check for derivative in a word context

Appendix Set B: Linearization

1. Compute L(x)=f(a)+f'(a)(x-a) for a smooth function
2. Use L(x) to approximate a root nearby and estimate error sign
3. Discuss radius of good approximation
4. Contrast linear vs secant approximation

Appendix Set C: Graph Features

1. From a graph, estimate f'(a) using symmetric difference
2. Identify where derivative is zero or undefined
3. Match slope fields to candidate functions
4. Explain concavity via the behavior of f'

Appendix Set D: Notation & Rigor

1. Translate Leibniz, Lagrange, and Newton notations
2. State domain restrictions before differentiating
3. Use abla notation contextually (concept only)
4. Differentiate between value, rule, and operator

Appendix Set E: Numerical Derivative

1. Compute forward/backward/central differences
2. Compare truncation vs rounding errors
3. Choose step h to balance accuracy and noise
4. Explain why central difference is often superior

Review Pack 1: Rules Mix

• Differentiate polynomial × exponential
• Use logarithmic differentiation for a power
• Chain rule on nested trig-composite
• Interpret slope in an application

Review Pack 2: Domains

• Mark where derivative undefined (division/log/roots)
• Left/right derivatives at a kink
• One-sided limits and piecewise rules
• Units on final numeric answers

Review Pack 3: Error Estimation

• Use linearization to estimate f(a+Δx)
• Bound the error using |f''| on an interval
• Report percentage error with units
• State conditions for validity

Challenge 1: Ternary Chain

1. Differentiate e^{sin(x^2)}
2. Evaluate at a special angle
3. Discuss small-angle approximation
4. Explain growth vs oscillation tradeoff

Challenge 2: Quotient Trap

1. Differentiate (xsin x)/(1+cos x)
2. Rationalize using trig identities
3. State excluded x-values
4. Describe sign of derivative near 0

Challenge 3: Implicit Curve

1. Given x^2+y^2=1+xy, find dy/dx
2. Find slope at intersection with x-axis
3. Explain symmetry arguments
4. Classify tangent behavior near (1,0)

Application 1: Population Growth

• P(t) = 1000e^(0.03t) models city population. Find dP/dt rate.
• Use chain rule: dP/dt = 1000 × 0.03 × e^(0.03t).
• Units: people per year. Positive means growing population.
• Linearization valid only for small time intervals around given point.

Application 2: Drug Concentration

• C(t) = 5te^(-0.2t) mg/L in bloodstream. Find dC/dt elimination rate.
• Use product rule: dC/dt = 5e^(-0.2t) + 5t(-0.2)e^(-0.2t).
• Units: mg/L per hour. Sign indicates increasing/decreasing concentration.
• Linear approximation breaks down as exponential decay dominates.

Application 3: Temperature Change

• T(t) = 20 + 15sin(πt/12) models daily temperature in °C.
• Chain rule: dT/dt = 15 × (π/12) × cos(πt/12).
• Units: °C per hour. Rate of temperature change throughout day.
• Linearization only accurate within 1-2 hours of reference time.

Application 4: Velocity Analysis

• s(t) = t³ - 6t² + 9t position function in meters.
• Velocity: v(t) = ds/dt = 3t² - 12t + 9 using power rule.
• Units: m/s. Sign indicates direction of motion.
• Linear approximation of position fails over long time intervals.

Application 5: Economic Marginal Cost

• C(x) = 0.01x³ - 0.5x² + 50x + 2000 total cost in dollars.
• Marginal cost: MC(x) = 0.03x² - x + 50 using polynomial rules.
• Units: dollars per unit. Cost to produce one additional item.
• Linear approximation valid only for small production changes.

Application 6: Related Rates

• Ladder sliding: x² + y² = 25², dx/dt = 3 ft/s given.
• Find dy/dt using implicit differentiation of constraint.
• Units: ft/s. Negative indicates downward motion.
• Model valid only while ladder maintains contact with wall.

Application 7: Fluid Dynamics

• V(t) = πr²h where r(t) changes. Find dV/dt flow rate.
• Product rule and chain rule: dV/dt = π[2r(dr/dt)h + r²(dh/dt)].
• Units: cubic units per time. Volume change rate.
• Linear approximation assumes constant rates over small intervals.

Application 8: Optimization Prep

• Surface area S(r) = 2πr² + 2πrh with constraint V = πr²h.
• Substitute h = V/(πr²): S(r) = 2πr² + 2V/r.
• Find dS/dr = 4πr - 2V/r² for critical points.
• Tangent line approximation fails far from optimal point.

Application 9: Compound Growth

• A(t) = 5000e^(rt) investment model. Find dA/dt growth rate.
• Chain rule: dA/dt = 5000r × e^(rt) where r is interest rate.
• Units: dollars per time. Instantaneous growth rate.
• Linear approximation underestimates exponential growth over time.

Application 10: Decay Processes

• N(t) = N₀e^(-λt) radioactive decay. Find dN/dt decay rate.
• Exponential rule: dN/dt = -λN₀e^(-λt) = -λN(t).
• Units: particles per time. Negative indicates loss.
• Linear approximation only valid for short time spans.

Drill Set 1

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 2

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 3

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 4

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 5

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 6

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 7

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 8

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 9

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 10

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 11

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 12

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 13

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 14

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 15

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 16

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 17

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 18

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 19

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 20

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 21

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 22

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 23

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Drill Set 24

1. Differentiate a piecewise-defined function at the junction (check limit of difference quotient).
2. Compute the derivative of $e^{x^2}sin x$ and simplify factors.
3. Use the derivative to estimate $f(a+Delta x)$ from $f(a)$ with a small $Delta x$.
4. Explain the meaning of negative derivative at a point in the given context.

Tangent as Limit of Secants

The secant slope between $(a,f(a))$ and $(a+h,f(a+h))$ is the average rate of change. As $h o 0$, secant lines approach the tangent line with slope $f'(a)$.

Tangent Line Equation

This linear approximation describes local behavior. For small $Delta x$, we have$;Delta y approx f'(a),Delta x$.

Power Rule

Valid for real n where expression is defined.

Constant Multiple

c is a constant.

Sum Rule

Linear operator.

Product Rule

Differentiate one factor at a time.

Quotient Rule

g(x) ≠ 0.

Chain Rule

Differentiate outer at inner, times inner derivative.

Trigonometric

Exponential & Logarithmic

Polynomial

Differentiate $f(x)=3x^4-2x^2+5$.

Chain Rule

Differentiate $y=sqrt{1+2x^3}$.

Product/Quotient

Differentiate $h(x)=x^2 e^x$ and $g(x)=dfrac{sin x}{x}$.

Tangent Line

Find tangent to $y=x^2$ at $x=2$.

Set A: Power & Product

1. Differentiate 3x^5-2x^3+x
2. Differentiate x^2 e^{2x}
3. Find tangent at x=1
4. State units if x is seconds and y meters

Set B: Chain & Quotient

1. Differentiate y=sqrt{1+2x^3}
2. Differentiate (ln x)/(x^2)
3. Compute derivative sign on (0,1)
4. Sketch local behavior

Set C: Trig Mix

1. Differentiate sin x - frac12sin 2x
2. Solve f'(x)=0 in [0,2pi]
3. Classify extrema via second derivative
4. Interpret on unit circle

Drill 1: Limit vs Rule

• Compute derivative at a via limit
• Confirm using rule-based differentiation
• Use derivative to approximate near a
• Explain units

Drill 2: Piecewise

• Check differentiability at the junction
• Left/right derivatives match?
• Linearize each side
• Contextual meaning