EC-18.3 The Derivative, and What It Is For

What this is and why it exists

The derivative has three readings, and different subjects reach for different ones without saying which.

As a slope it describes a graph. As a rate it describes how fast a physical quantity is changing. As a sign it says whether that quantity is rising or falling. A reader who has only the slope picture is missing most of what the derivative is used for in this area.

Current is the derivative of charge with respect to time. Velocity is the derivative of position. The small-signal behaviour of a device is the derivative of its characteristic curve at the operating point. Each of those is a rate, and none of them is a picture of a graph.

The vocabulary

• Derivative — the limit of the change in output divided by the change in input, as the change in input goes to zero.
• Rate of change — the same quantity read physically.
• Chain rule — the rule for differentiating a composition of functions.
• Implicit differentiation — differentiating a relation that has not been rearranged to give one quantity in terms of the other.
• Related rates — two changing quantities tied together by a fixed relation.
• Mean value theorem — over an interval, the average rate of change is achieved at some position inside it.
• Stationary position — a position where the derivative is zero.
• Second derivative — the derivative of the derivative, describing curvature.
• Newton's method — using the derivative to step towards a position where an expression is zero.

The mental model

One definition, three readings, and it is worth being deliberate about moving between them.

The definition is a limit. Take the change in output over a small change in input, and let the change in input go to zero. What survives is the instantaneous rate. Every use of the derivative in this area is that number under a different name.

The rules are next, and there are four. The derivative of a sum is the sum of the derivatives. The product rule handles a product, the quotient rule a ratio. The chain rule handles a composition, and it is the one you will use more than the other three combined.

The chain rule deserves its own sentence. It says that a change carried through a sequence of dependencies multiplies at each stage. Temperature moves the sensor voltage. The sensor voltage moves the amplifier output. To find how the output responds to temperature, multiply the two rates. That reading makes the rule memorable in a way the notation does not.

Implicit differentiation follows from the chain rule and looks harder than it is. Some relations cannot be rearranged to give one quantity alone. Differentiating them as they stand still works, provided every appearance of the dependent quantity is differentiated with the chain rule attached. The same technique answers related-rate questions, where two quantities move together under a fixed relation.

The mean value theorem is the quiet engine of the topic. It says the average rate of change over an interval is achieved at some instant inside it. From that one statement come several results you would otherwise have to take on faith. A function whose derivative is zero everywhere on an interval is constant there. A function with positive derivative is increasing. Those are used constantly and are almost never justified.

Then optimisation, which is the reason engineers care. A maximum or a minimum in the interior of an interval has zero derivative, because a non-zero rate means you could do better by moving. The second derivative says which kind of position you found: negative for a peak, positive for a valley, and no verdict when it is zero.

The mistake to guard against is forgetting the ends. A quantity on a closed interval can take its largest value at an endpoint, where the derivative need not be zero at all. Component values, supply voltages and operating temperatures all come with limits, so the endpoint case is the normal case in engineering rather than the exception.

Newton's method closes the topic and connects it to the last one. Bisection needed only continuity and converged slowly. Newton uses the derivative to follow the tangent line to where it crosses zero, and takes that crossing as the next guess. It converges far faster when it converges. It can also run away entirely when the starting guess is poor or the derivative is near zero, which is the price of the speed.

What you should now be able to explain or do

• Move between the derivative as a slope, as a rate and as a sign, and say which a given problem wants.
• Differentiate any expression built from the standard families using the four rules.
• Use the chain rule to carry a change through a sequence of physical dependencies.
• Differentiate a relation implicitly and answer a related-rate question.
• Find a maximum or minimum on a closed interval, including the endpoints.
• Explain when Newton's method beats bisection, and when it fails.

Check yourself