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Math for ME · Chapter 4 of 19 · Beginner

Single-Variable Calculus: Derivatives

Velocity, acceleration, sensitivity, and optimization are all the same question: how fast does this change?

The thread: You can describe where things are. The deeper engineering question is how fast they change, and that instantaneous rate is the derivative.

Readiness check

From Algebra, Functions, and Engineering Notation plus Trigonometry and Geometry. Tick only what you can do closed-notes.

Compute the slope of a straight line from two points.
Expand and simplify polynomial expressions.
Evaluate exponential and trig functions at given points.
Sketch the five function shapes from Algebra and Functions.
Read units like m/s and m/s² as "per second" rates.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsReview function shapes and slopes in Algebra, Functions, and Engineering Notation first.

3 or more weak itemsStep back to Algebra, Functions, and Engineering Notation and Trigonometry and Geometry.

The core idea

The derivative is the slope of a curve at a point: the instantaneous rate of change.

v = ds/dta = dv/dt

A handful of rules generate every derivative you will need: power, chain, product, quotient, plus the tables for e^x, ln x, sin, cos. The chain rule carries the most weight in engineering, because real formulas are functions of functions.

The skill works when: you read a derivative as a physical rate with units (m/s, Pa/K, A/s), not as a symbol game.

The skill breaks down when: the chain rule's inner derivative is forgotten, or a maximum is claimed without checking the derivative actually changes sign.

The concept. Zoom in far enough and every smooth curve is a straight line. That local slope is the derivative: the speedometer reading, not the trip average.

The skills, taught in order

4.1 The derivative as limit, slope, and rate

The derivative is the limit of the average rate of change as the interval shrinks to zero. Geometrically it is the slope of the tangent line; physically it is a rate, with units of output per input.

f′(x) = lim_h→0 [f(x+h) − f(x)]/h

You rarely evaluate this limit by hand. Its value is conceptual: it is what every rule below is a shortcut for.

4.2 The rules that generate every derivative

Four rules and one small table cover almost all engineering work.

Rule	Form
Power	d/dx (xⁿ) = n xⁿ⁻¹
Product	(uv)′ = u′v + uv′
Quotient	(u/v)′ = (u′v − uv′)/v²
Chain	d/dx f(g(x)) = f′(g)·g′(x)

The standard table of functions worth memorising (x in radians):

f(x)	e^x	ln x	sin x	cos x
f′(x)	e^x	1/x	cos x	−sin x

4.3 The chain rule is the engineering workhorse

Real formulas are functions of functions, so the chain rule appears constantly. To differentiate e^−t/τ, the outer derivative is e^−t/τ and the inner derivative of −t/τ is −1/τ, giving −(1/τ)e^−t/τ. The most common slip is stopping at the outer derivative and forgetting the inner factor.

4.4 Higher derivatives and optimisation

Differentiating again gives acceleration (from velocity) and concavity (from a curve). To find a maximum or minimum, set f′(x) = 0 to locate candidates, then confirm with the sign change or the second derivative: f″ < 0 at a maximum, f″ > 0 at a minimum. Solving f′ = 0 on its own proves nothing is optimal.

4.5 Linearisation: the engineer's estimate

Near a known point a curve is almost its own tangent line, so a small change in input produces a predictable change in output:

Δy ≈ (dy/dx) Δx

This is how tolerances and sensitivities are estimated without re-solving the whole model. If a quantity scales as Lⁿ, a small fractional change in L produces about n times that fractional change in the quantity.

Engineering connection: Dynamics, Controls, Thermodynamics, Fluid Mechanics.

Worked example: from position to velocity and acceleration

A slider moves along a rail with position s(t) = 4t³ − 6t² metres (t in seconds). Find its velocity and acceleration at t = 2 s, and find when the slider is momentarily at rest.

Figure 1. The position curve: the slider backs up, stops at t = 1 s, then accelerates forward. At t = 2 s the tangent slope is 24 m/s.

ProblemFind v(2), a(2), and the rest instants for the motion in Figure 1.
Given / finds(t) = 4t³ − 6t² m. Find v(t), a(t), v(2), a(2), and times where v = 0.
AssumptionsStraight-line motion; s differentiable (smooth mechanism).
ModelVelocity is the first derivative of position; acceleration the second.
Equationsv = ds/dt = 12t² − 12t a = dv/dt = 24t − 12
Solvev(2) = 12(4) − 12(2) = 24 m/s. a(2) = 24(2) − 12 = 36 m/s². Rest: 12t(t − 1) = 0, so t = 0 and t = 1 s.
CheckUnits: each differentiation divides by seconds: m, then m/s, then m/s². Numerically: s(2.01) − s(2) = 8.2424 − 8 = 0.2424 m over 0.01 s gives 24.2 m/s, matching v(2) ≈ 24.
ConclusionThe slider reverses at t = 1 s (v changes sign), which a designer must know for impact and clearance. The derivative did not just compute a number: it found the event.

Result. v(2) = 24 m/s, a(2) = 36 m/s²; the slider is at rest at t = 0 and t = 1 s.

04b

Worked example 2: how a length error grows in deflection

A cantilever beam's tip deflection is δ = PL³/(3EI). During manufacture the length L comes out 4% longer than the nominal value. Estimate the percentage change in deflection using linearisation, without recomputing δ from scratch.

Given / findδ = PL³/(3EI); ΔL/L = +0.04. Find the approximate Δδ/δ.
ModelOnly L changes, so treat δ as a function of L and use Δδ ≈ (dδ/dL) ΔL.
Differentiatedδ/dL = 3PL²/(3EI) = PL²/(EI). This equals 3δ/L, since δ = PL³/(3EI).
Form the fractional changeΔδ/δ ≈ (dδ/dL)(ΔL)/δ = (3δ/L)(ΔL)/δ = 3 (ΔL/L).
EvaluateΔδ/δ ≈ 3 × 0.04 = 0.12, a 12% increase in deflection.
CheckExact: (1.04)³ = 1.125, a 12.5% increase. The linear estimate of 12% is close, and slightly low as expected because the curve bends upward.
ConclusionBecause deflection scales with L³, a length error is amplified roughly three-fold. That factor, read straight off the exponent, is why a designer controls length tightly. This is the everyday use of derivatives in tolerance work.

Result. A 4% length error gives about a 12% deflection increase (exact 12.5%).

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Forgetting the inner derivative	d/dt of e^−0.1t reported as e^−0.1t	"What is the derivative of the inside function?"	Chain rule: outer derivative times inner derivative. Here: −0.1 e^−0.1t.
Slope confused with value	"v is biggest where s is biggest"	"Am I reading the height of the curve or its steepness?"	The derivative is steepness. A curve can be high and flat (v = 0 at the peak).
Claiming a maximum from f' = 0 alone	Minimum or saddle reported as the optimum	"Does the derivative change from + to − here?"	Check the sign change or the second derivative before announcing an optimum.
Degrees inside trig derivatives	d/dx sin x "= cos x" failing numerically	"Is x in radians?"	The clean rules (sin → cos) hold only in radians. Convert before differentiating.

Practice ladder

Level 1 · Direct skill

Differentiate y = 5x³ − 2x² + 7, and evaluate the slope at x = 2.

Show answer

dy/dx = 15x² − 4x; at x = 2: 60 − 8 = 52.

Then differentiate y = (2x + 1)⁴ with the chain rule.

Show answer

dy/dx = 4(2x + 1)³ · 2 = 8(2x + 1)³. The inner derivative is the 2; forgetting it is the classic slip.

Level 2 · Mixed concept

A part cools as T(t) = 80e^−0.1t °C above ambient (t in minutes). How fast is it cooling at t = 10 min?

Show answer

dT/dt = −8e^−0.1t; at t = 10: −8e⁻¹ = −2.94 °C/min. Chain rule with units attached.

Differentiate P(t) = t² sin t with the product rule.

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P′ = 2t sin t + t² cos t. Each factor takes a turn being differentiated while the other is held fixed.

Level 3 · Independent problem

A drive produces force F = 500 − 20v newtons at speed v m/s, so the power is P = Fv = 500v − 20v². Find the speed for maximum power and the power there. Verify it is a maximum.

Show answer

dP/dv = 500 − 40v = 0 gives v = 12.5 m/s; P = 500(12.5) − 20(156.25) = 3125 W. Second derivative −40 < 0: a maximum. This pattern (peak power at half the no-load force point) recurs in motor selection.

Level 4 · Transfer to real engineering

Take any formula with one design variable (pendulum period, pipe pressure drop, beam deflection). Differentiate it with respect to that variable and use Δy ≈ (dy/dx)Δx to predict the effect of a 5% manufacturing error. State which variable the design is most sensitive to and why.

What good work looks like

A correct derivative with units, a numeric 5%-error estimate, and a sensitivity statement such as "deflection scales with L³, so a 5% length error gives roughly a 15% deflection error."

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Here is my derivative, step by step. Tell me which line breaks a rule, and which rule, but not the fix."

"Generate five chain-rule functions of increasing nesting. Grade each of my answers before showing the next."

"Differentiate this for me." The rules must be reflexive before Dynamics.

"Where is the maximum?" Setting f' = 0 and verifying the sign change is the entire optimization skill.

Portfolio task

Build a one-page "Rates Sheet" for one real system you own (bike, kettle, phone battery): measure or estimate a quantity at several times, fit a rough curve, estimate the derivative at two points numerically, and interpret both rates physically.

Must include: the data table, the two numerical slopes with units, and one sentence on where the rate is largest and why that makes physical sense.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Define the derivative physically and geometrically.

Physically: the instantaneous rate of change with units of output per input. Geometrically: the slope of the tangent line.

2. State the chain rule and give an engineering example.

d/dx f(g(x)) = f'(g)·g'. Example: d/dt e^−t/τ = −(1/τ)e^−t/τ, the decay rate of every first-order system.

3. What are the derivatives of sin x, cos x, e^x, ln x?

cos x; −sin x; e^x; 1/x. Valid with x in radians.

4. What does the second derivative tell you about motion and about curves?

Motion: acceleration. Curves: concavity, and the max/min test (negative at a maximum).

5. Write the linearization formula and its use.

Δy ≈ (dy/dx)·Δx: predict the effect of small changes and tolerances without re-solving the whole model.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-solve the slider example, including the numeric check.

+3 daysOne optimization (Level 3 style) with your own numbers.

+7 daysMixed set: a derivative, a triangle, and a rearrangement.

+30 daysUse v = ds/dt inside a Dynamics kinematics problem.

Textbook mapping

Item	Mapping
Main sources	Stewart, Calculus: Early Transcendentals (limits and derivatives chapters)
Core topics	4.1 Limits as approximations · 4.2 Slope · 4.3 Rate of change · 4.4 Rules · 4.5 Chain rule · 4.6 Product and quotient · 4.7 Standard table · 4.8 Higher derivatives · 4.9 Linearization
Engineering connection	Dynamics (velocity, acceleration), Controls (rates), Thermodynamics and Fluids (gradients in time).
Skip on first pass	Formal limit proofs, implicit differentiation beyond basics, related-rates puzzles without engineering content.
Read next	Integrals.