The derivative

The derivative is the most useful single idea in engineering mathematics. It measures how fast one quantity changes in response to another: how quickly a beam deflects as load increases, how fast a motor accelerates, how rapidly the temperature of a reactor climbs. Velocity, acceleration, current, power, strain rate and heat flux are all derivatives.

In Lesson 1 you learned what a limit is. In this lesson you will use limits to define the derivative precisely, compute derivatives directly from the definition, interpret them as slopes and as rates, write equations of tangent lines, and see when a derivative fails to exist. Lesson 3 then gives you shortcut rules so you rarely need the definition in practice. Understanding where the rules come from will make you far better at using them.

From average rate to instantaneous rate

Start with something you already know. If a quantity changes from \(f(a)\) to \(f(b)\) as the input goes from \(a\) to \(b\), its average rate of change is:

If \(f\) is position in meters and \(x\) is time in seconds, this is average velocity in m/s. If \(f\) is the volume in a tank and \(x\) is time, it is the average fill rate.

Averages hide detail. A car that averages 60 km/h over an hour may have been stopped at lights and briefly doing 90. What we often need is the rate at one instant. The trick is to compute the average over shorter and shorter intervals and see what value they approach.

That limiting value is the instantaneous rate of change: the derivative.

Secant lines become tangent lines

The average rate of change has a picture. On the graph of \(f\), it is the slope of the secant line through \((a, f(a))\) and \((a + h, f(a + h))\). As \(h\) shrinks, the second point slides along the curve toward the first, and the secant line pivots toward a limiting position: the tangent line.

So the derivative has two equivalent meanings, and it helps to keep both in mind:

• Geometrically, it is the slope of the tangent line to the graph.
• Physically, it is the instantaneous rate of change of the output with respect to the input.

The definition of the derivative

The derivative of \(f\) at \(x = a\) is

provided the limit exists. An equivalent form, useful for some problems, lets \(x\) approach \(a\):

Notice that substituting \(h = 0\) always gives \(0/0\). Every derivative is an indeterminate-form limit, which is why Lesson 1 spent so much time on them. The algebra techniques from that lesson (expand and cancel, rationalize, combine fractions) are exactly what you need here.

These three results are all special cases of the power rule you will meet in Lesson 3: \(\dfrac{d}{dx}x^n = nx^{n-1}\), with \(n = 2\), \(n = -1\) and \(n = \tfrac12\).

Notation and units

Several notations are in common use, and you need to read all of them:

• Prime notation: \(f'(x)\), or \(y'\) when \(y = f(x)\).
• Leibniz notation: \(\dfrac{dy}{dx}\) or \(\dfrac{d}{dx}f(x)\). It reminds you that the derivative is a limit of \(\Delta y / \Delta x\).
• Dot notation: \(\dot{x}\) means \(dx/dt\), a derivative with respect to time. It is standard in dynamics and control.

Leibniz notation makes units obvious. The units of \(dy/dx\) are the units of \(y\) divided by the units of \(x\).

• Position (m) with respect to time (s): velocity in m/s.
• Velocity (m/s) with respect to time: acceleration in m/s².
• Charge (C) with respect to time: current in C/s, which is amperes.
• Energy (J) with respect to time: power in J/s, which is watts.
• Volume (m³) with respect to time: flow rate in m³/s.

The tangent line and linear approximation

Once you know the slope \(f'(a)\) and the point \((a, f(a))\), the tangent line follows from the point-slope form of a line:

Close to the point of tangency, the tangent line and the curve are nearly identical. That makes the tangent line a cheap, accurate approximation, called the linear approximation or linearization:

Linearization is everywhere in engineering. Control engineers linearize nonlinear systems around an operating point so they can use linear design methods. Structural engineers use small-deflection theory, which is a linearization. The small-angle approximation \(\sin\theta \approx \theta\) is the linearization of sine at 0.

The derivative as a function

If you compute \(f'(x)\) at every point where it exists, you get a new function, \(f'\). Its graph tells you about the shape of the graph of \(f\):

• Where \(f'(x) > 0\), the tangent slopes upward, so \(f\) is increasing.
• Where \(f'(x) < 0\), \(f\) is decreasing.
• Where \(f'(x) = 0\), the tangent is horizontal. This often marks a peak or a valley.

Reading the graph of \(f'\) from the graph of \(f\), and the other way round, is a skill worth practicing. Look at where \(f\) is flat (\(f' = 0\)), where it climbs steeply (\(f'\) large and positive) and where it falls (\(f'\) negative).

Lesson 4 turns this into a systematic method for finding maximum and minimum values, which is the basis of optimization.

When a derivative does not exist

A function is differentiable at \(a\) when \(f'(a)\) exists. Differentiability is a stronger condition than continuity:

• If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).
• The reverse is false. A function can be continuous at a point and still have no derivative there.

There are four common ways for a derivative to fail to exist:

• A corner: the slopes from the left and right differ. For \(|x|\) at 0, the left slope is −1 and the right slope is +1.
• A cusp: the slopes from each side become infinite in opposite directions, as with \(x^{2/3}\) at 0.
• A vertical tangent: the slope becomes infinite, as with \(x^{1/3}\) at 0.
• A discontinuity: if the function jumps or has a hole, there is no tangent line at all.

Corners appear in real engineering models wherever behavior switches abruptly: a cable going slack, a contact closing, a material yielding. Models with corners need extra care in analysis and simulation.

Higher derivatives

The derivative of a derivative is the second derivative, written \(f''(x)\), \(\dfrac{d^2y}{dx^2}\) or \(\ddot{x}\) for time derivatives. You can keep going to third and higher derivatives.

In motion, if \(s(t)\) is position, then \(v = \dot{s}\) is velocity, \(a = \dot{v} = \ddot{s}\) is acceleration, and the derivative of acceleration is jerk. Elevator and rail designers limit jerk because sudden changes in acceleration feel uncomfortable.

The second derivative also describes curvature. Where \(f''(x) > 0\), the slope is increasing and the graph bends upward (concave up, like a cup). Where \(f''(x) < 0\), it bends downward (concave down, like a cap).

Higher derivatives appear throughout structural engineering. For a beam with deflection \(y(x)\), the slope of the beam is \(y'\), the bending moment is proportional to \(y''\), the shear force to \(y'''\), and the distributed load to \(y''''\) (with signs depending on the convention used). This is the Euler–Bernoulli beam equation, \(EI\,y'''' = w(x)\), which you will meet in Mechanics of Solids.

Estimating derivatives from data

In practice, engineers often have measurements rather than formulas: a temperature logged every ten seconds, a displacement sensor read at 1 kHz, a survey of a road profile every 20 meters. The definition of the derivative still tells you what to do. Replace the limit with a small but finite step \(h\), the spacing of your data.

Three finite-difference formulas are in everyday use:

The forward and backward formulas are simply the difference quotient with \(h\) fixed. The central difference averages them, using one point on each side. It is usually far more accurate, because the errors of the forward and backward versions largely cancel.

There is a catch. Making \(h\) smaller reduces the approximation error, but it amplifies measurement noise: a small error in each reading becomes a large error when divided by a tiny \(h\). Noisy data is usually smoothed or fitted with a curve before it is differentiated. This trade-off appears in every control system that uses a derivative term.