Applications of the derivative
The derivative finds maxima and minima, justifies linear approximations and solves the optimization problems that appear in every design.
01Increase, decrease and extrema
The sign of the derivative tells you where the function is heading, and the points where it vanishes are the candidates for maximum or minimum:
| If on an interval… | then f… |
|---|---|
| f′(x) > 0 | increases |
| f′(x) < 0 | decreases |
| f′(c) = 0 or does not exist | has a critical point at c: it has to be analyzed |
The first-derivative test settles each critical point by looking at how the sign of f′ changes as it passes through it: from + to − is a maximum; from − to + is a minimum; if it does not change, it is not an extremum (like x³ at the origin).
f(x) = x³ has f′(0) = 0 and at 0 there is neither a maximum nor a minimum: there is an inflection point with a horizontal tangent. That is why a critical point is a candidate, not a conclusion.
To find the absolute extrema on a closed interval [a, b] —which exist, by Weierstrass— you compare the values at the critical points and at the two endpoints of the interval. The largest and the smallest of that list are the answer.
02Concavity and inflection
The second derivative measures how the slope changes: if f″ > 0 the curve is concave up and the slope increases; if f″ < 0, concave down. Where f″ changes sign there is an inflection point.
From this comes the second-derivative test, quicker when it can be applied: if f′(c) = 0 and f″(c) > 0, there is a minimum at c; if f″(c) < 0, a maximum. If f″(c) = 0, the test is inconclusive and you have to go back to the first-derivative test.
The marked points are the critical and inflection points, computed live by checking where each derivative changes sign.
03Rolle, the mean value theorem and L’Hôpital
If f is continuous on [a, b] and differentiable on (a, b), there is at least one point c where the slope of the tangent equals the slope of the secant joining the endpoints:
Translated: if a car covers 120 km in one hour, at some instant its speedometer read exactly 120 km/h. Rolle’s theorem is the special case with f(a) = f(b), where that slope is zero.
Consequences of Lagrange’s theorem are used without being named: a function with zero derivative on an interval is constant, and two functions with the same derivative differ by a constant —the basis of the indefinite integral— and the increase and decrease criteria of section 1.
For a 0/0 or ∞/∞ indeterminate form, the limit of the quotient equals the limit of the quotient of the derivatives, if the latter exists:
It can be repeated if the indeterminate form persists. It is not the derivative of the quotient: it is the two derivatives taken separately. And you have to check that the indeterminate form is present before applying it.
04Optimization: the design problem
Optimizing is the bread and butter of engineering: maximizing power, efficiency or bandwidth; minimizing consumption, cost or dissipation. The method is always the same:
- Write the quantity to optimize as a function of a single variable, using the relations of the problem to eliminate the others.
- Establish the domain that makes physical sense.
- Differentiate, set equal to zero and solve.
- Check that it is a maximum or a minimum, and also check the boundaries of the domain.
A voltage source V with internal resistance r drives a load R. The current is i = V/(r + R) and the power in the load is:
Differentiating with the quotient rule:
P′ vanishes at R = r, and since P′ > 0 before and P′ < 0 after, it is a maximum. The maximum power is P = V²/(4r), and the efficiency at that point is barely 50%: half of the energy is dissipated inside the source. That is why the criterion is used with small signals —an antenna, a radio-frequency amplifier— and not in power distribution, where what matters is efficiency.
05Taylor polynomials
The tangent line approximates a function with a polynomial of degree 1. By adding terms with the successive derivatives, the approximation improves:
The two approximations most used in electronics come from here: sin x ≈ x for small angles and ex ≈ 1 + x, which linearizes the diode equation around an operating point.
06Where all this is used
- Operating point and maximum swing. Placing a transistor’s Q point in the middle of the load line is an optimization problem: maximizing the symmetric swing of the signal before clipping.
- Impedance matching. The result R = r from section 4 is the DC version of the maximum transfer criterion, which at radio frequency is solved with the Smith chart.
- Harmonic distortion. When you expand an amplifier’s nonlinear characteristic in a Taylor series, vo = a₁v + a₂v² + a₃v³ + …, the quadratic term generates the second harmonic and the cubic one the third. Distortion is not an accident: it is the higher-order terms of the expansion.
- Newton–Raphson. The numerical method that solves nonlinear equations —the operating point of a diode, for example— is “moving along the tangent”: xn+1 = xn − f(xn)/f′(xn). It converges much faster than bisection.
- Sensitivity and tolerances. The derivative of the output with respect to each component tells you how much its tolerance affects the result, and where it is worth spending on precision.
07In the lab
With a 10 V source and 100 Ω in series —the simulated internal resistance—, measure the power in loads of 10, 50, 100, 200 and 500 Ω. Plot P as a function of R and check that the maximum is at 100 Ω and equals V²/4r = 250 mW.
Drive an amplifier close to clipping and observe the spectrum with an analyzer. Relate the appearance of harmonics to the nonlinear terms of the Taylor expansion. You can practice this in the instrument simulator.
Set up the junction temperature of a transistor as a function of the thermal resistance of the heat sink and of cost, and find the minimum cost that meets the thermal specification. It is the same four-step method.
08Common mistakes
- Taking every critical point to be an extremum without analyzing the sign change.
- Forgetting the endpoints of the interval when looking for absolute extrema.
- Applying L’Hôpital without an indeterminate form, or differentiating the quotient instead of each part.
- Optimizing without a physical domain: a negative resistance can be a mathematical solution and not exist.
- Confusing maximum power with maximum efficiency. At R = r the efficiency is 50%.
- Using sin x ≈ x outside small angles: at 30° the error is already almost 5%.
09Self-assessment
Find and classify the extrema of f(x) = x³ − 3x.
f′ = 3x² − 3 = 0 at x = ±1. f″ = 6x: at x = 1 it is positive (minimum, f = −2) and at x = −1 it is negative (maximum, f = 2).
Where is the inflection point of f(x) = x³ − 3x?
f″ = 6x vanishes and changes sign at x = 0: that is where the inflection is.
Compute limx→0 (ex − 1)/x with L’Hôpital.
It is 0/0; differentiating top and bottom leaves ex/1 → 1.
A 12 V source with r = 50 Ω. What load draws the maximum power and how much is it?
R = 50 Ω and P = V²/(4r) = 144/200 = 0.72 W.
Write the degree-3 Maclaurin polynomial of sin x.
x − x³/6. The degree-2 term is zero because the sine is odd.
What does the mean value theorem say about a signal that goes from 0 to 5 V in 1 µs?
That at some instant its slope was exactly 5 V/µs. If the amplifier has a lower slew rate, it cannot have produced that edge.
10Further reading
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 4 is exactly this topic: extrema, the mean value theorem, L’Hôpital’s rule and optimization.
- Louis Leithold. The Calculus 7. 7th ed. (Spanish edition, “El cálculo”), Oxford University Press. Many solved optimization problems, with the setup detailed step by step.
- Tom M. Apostol. Calculus, Volume 1. 2nd ed. (Spanish edition, “Calculus, volumen 1”), Reverté. Taylor’s formula with the remainder proved, so you know how good the approximation is.