Differentiable functions
The derivative measures how fast something changes. In a circuit it is the current through a capacitor, the voltage across an inductor and the response speed of an amplifier.
01The slope at a point
The slope of a line is computed from two points. What about the slope of a curve, at a single point? The way out is to take two very close points, compute the slope of the line joining them —the secant— and see what value it approaches as the points come together. That limit is the derivative:
Geometric: the slope of the tangent line.
Physical: the instantaneous rate of change: if x(t) is position, x′(t) is velocity.
Engineering: sensitivity. How much the output changes for each unit the input changes,
right at that operating point.
If the limit exists, the function is differentiable at a. Every differentiable function is continuous, but the converse is false: f(x) = |x| is continuous at 0 and not differentiable, because from the left the slope is −1 and from the right +1. On a graph, the non-differentiable points are corners, vertical tangents and discontinuities.
Shrink h and watch the secant lie down onto the tangent. The curve below is the derivative, point by point.
02The rules, which save you from taking the limit every time
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| k (constant) | 0 | sin x | cos x |
| xn | \( n \, x^{n-1} \) | cos x | −sin x |
| ex | ex | tan x | \( \sec^2 x = 1 + \operatorname{tan}^2 x \) |
| ax | \( a^x \ln a \) | arctan x | \( \dfrac{1}{1 + x^{2}} \) |
| ln x | \( \dfrac{1}{x} \) | \( \sqrt x \) | \( \dfrac{1}{2\sqrt x} \) |
And the four combination rules:
For a composition, the derivatives are multiplied: (g ∘ f)′(x) = g′(f(x)) · f′(x). You differentiate the outer function evaluated at the inner one, and multiply by the derivative of the inner function.
Example: the derivative of e−t/RC is e−t/RC · (−1/RC). That factor that appears “from the inside” is what makes the discharge current of a capacitor depend on the time constant.
03Derivatives used in electronics
If v(t) = V·sin(ωt), then i = C·dv/dt = C·V·ω·cos(ωt).
Three conclusions from a single derivative: the current leads by 90° the voltage (the cosine leads the sine), its amplitude is proportional to the frequency —hence capacitive reactance is 1/(ωC)— and so a capacitor “lets through” fast signals better.
For a sinusoidal output vo = A·sin(2πft), the maximum slope is 2πfA, and it occurs at the zero crossing. An op amp with slew rate SR can follow it only if 2πfA ≤ SR.
With SR = 0.5 V/µs and A = 10 V, the maximum undistorted frequency is f = SR/(2πA) = 500,000/(2π·10) ≈ 7.96 kHz. Above that, the sine comes out triangular.
The second derivative, f″, is the derivative of the derivative: it measures how the rate of change changes. In mechanics it is acceleration; on a graph, the concavity.
04The differential: the best linear approximation
This is linearizing, and it is done all the time in electronics: the small-signal model of a transistor is the tangent line to its characteristic at the operating point. The dynamic resistance of a diode, rd = nVT/I, comes from differentiating its exponential equation and keeping the slope at the Q point. All the linear analysis of a small-signal amplifier rests on that approximation.
The differential is also useful for propagating errors: if a quantity is computed from another one measured with some uncertainty, the error is multiplied by the derivative.
05In the lab
Record the charging of a capacitor with the oscilloscope, export the points and compute the numerical derivative (v[i+1] − v[i])/Δt. Compare it with the current measured across a series resistor: they must be proportional, with constant C.
Drive an op amp with a square wave and measure the maximum slope of the output. Use it to compute the maximum frequency for a sine wave of 10 V peak, and check it by raising the frequency until the output turns triangular.
Measure the i-v curve of a diode point by point, plot it and estimate the slope at three different operating points. Compare with rd = 26 mV / I and discuss the agreement.
06Common mistakes
- Differentiating a product as the product of the derivatives.
- Forgetting the chain-rule factor, which is the costliest mistake and the quietest one.
- Confusing f′(x) with f(x′) or with the derivative evaluated at another point.
- Believing that continuous implies differentiable: the absolute value is the counterexample.
- Differentiating sin x with x in degrees: the formulas hold in radians.
- Using the small-signal model far from the operating point, where the tangent no longer approximates anything.
07Self-assessment
Differentiate f(x) = 3x⁴ − 5x + 2.
f′(x) = 12x³ − 5.
Differentiate f(t) = e−t/τ.
By the chain rule: f′(t) = −(1/τ)·e−t/τ.
Differentiate f(x) = x²·sin x.
Product rule: f′(x) = 2x·sin x + x²·cos x.
Why is |x| not differentiable at 0?
Because the difference quotient tends to −1 from the left and to +1 from the right: the limit does not exist. There is a corner.
A 100 nF capacitor has v(t) = 5·sin(2π·1000·t). What is the peak current?
i = C·dv/dt has a peak of C·V·ω = 100·10−9 · 5 · 2π·1000 ≈ 3.14 mA.
What does it mean for the derivative to be negative at a point?
That the function is decreasing there: if the input increases a little, the output goes down.
08Further reading
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. Chapters 2 and 3 take you from the difference quotient to all the rules, with physical applications.
- Hebe Rabuffetti. Introducción al análisis matemático (Cálculo 1) (in Spanish). El Ateneo. Differentiability and continuity treated with care, and many differentiation exercises.
- Tom M. Apostol. Calculus, Volume 1. 2nd ed. (Spanish edition, “Calculus, volumen 1”), Reverté. For the rigorous approach, with complete proofs of each rule.