EC-18.3 The Derivative, and What It Is For
You can differentiate the functions engineering actually uses, read a derivative as a rate of change, and find the value of a quantity that makes a design as good as it can be.
Before:nothing requiredUnlocks:EC-3. Differential Equations
Three readings of the same object are taught together on purpose: the derivative as a slope, as a rate, and as the number that decides whether a quantity is going up or down. Later subjects use all three without saying which one they mean. Current is the derivative of charge, velocity is the derivative of position, and the small-signal behaviour of a device is the derivative of its characteristic curve, so a reader who only has the slope picture is missing most of what the derivative is used for here.
Work through these
The derivative as a limit, as a slope, and as a rate of change
One definition supports three readings, and different subjects reach for different ones. Recognising that the slope of a graph and the rate at which a physical quantity changes are the same object is what makes the later applications legible.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseThe rules: sum, product, quotient and the chain rule
Four rules cover every expression built from the standard families, and the chain rule is the one that carries a change through a sequence of dependencies. It is used more heavily than the other three combined once several variables arrive.
MIT 18.01SC: Single Variable Calculus · CourseImplicit differentiation, and rates that depend on one another
Some relations cannot be rearranged to give one quantity in terms of the other, and differentiating them as they stand still works. The same technique answers questions where two changing quantities are tied together by a fixed relation.
The mean value theorem, and what it lets you conclude
Over an interval, the average rate of change is achieved at some instant inside it. That statement is the justification for most of what comes next, including the fact that a function with zero derivative everywhere is constant.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · CourseMaxima and minima, and the second derivative test
A turning point has zero derivative, and the sign of the second derivative says whether it is a peak or a valley. Endpoints have to be checked separately, which is where answers are most often lost.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseOptimisation in an engineering setting, and Newton's method for a root
Choosing a component value that maximises efficiency or minimises error is an optimisation problem written in ordinary words. Newton's method uses the derivative to find where an expression is zero, and it converges far faster than halving an interval.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · Course
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