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Differentiation
The rate of change at one instant. The limit, the derivative as the slope of a tangent, the rules for powers, products, sines, exponentials and functions of functions, the second derivative, maxima and minima, and related rates.
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Why this chapter
Why this chapter exists
Physics is about change. A velocity is how fast a position changes, a current how fast a charge flows, a force how fast a momentum changes. Each of these is a rate at one instant, and a rate at an instant is a derivative. This chapter builds the derivative from nothing and gives the handful of rules that the whole book uses. Then it applies them to the longest throw, a moving image and the falling top of a ladder.
You will learn
By the end of this chapter you will
- Explain the derivative as the limit of an average rate of change, and find one from a formula by the limit.
- Read a derivative as the slope of a tangent, and differentiate powers, sums, products, sines, exponentials and functions of functions.
- Find an acceleration with the second derivative, and read its sign from the bend of a graph.
- Find the greatest or least value of a physical quantity, and relate the rates of quantities tied by an equation.
Before you begin
Before you begin
You should be able to work with powers such as \(x^3\) and \(x^{-1}\), expand \((a + b)^2\), and read a value from a graph. Lesson 3 uses the sine and cosine of an angle, and the meaning of \(e^x\) and \(\ln x\). The Toolbox chapters on trigonometry and on exponentials cover them. No calculus is assumed.
Lessons
1
The Derivative as a Limit
How a rate over an interval becomes a rate at one instant. The average rate of change, the limit, the derivative \(dy/dx\) and what its letter d means.
Average rate of changeThe derivative as a limit
2
The Slope of the Tangent and the Power Rule
The derivative is the slope of the tangent to a graph. Its sign says whether a quantity rises or falls. The rule for powers of \(x\), for constants and sums, and the product and quotient rules.
The derivative is the slope of the tangentDerivatives of powers and sumsThe product and quotient rules
3
Sines, Exponentials and the Chain Rule
The derivatives of \(\sin x\), \(\cos x\), \(e^x\) and \(\ln x\), and the chain rule for a function of a function, which turns \(\sin\omega t\) into \(\omega\cos\omega t\).
Derivatives of sin cos and expThe chain rule
4
The Second Derivative
Differentiating twice. The second derivative \(d^2x/dt^2\) is the acceleration, and its sign is the way a graph bends: up like a cup, or down like a dome.
The second derivativeConcavity and the bend of a graph
5
Maxima and Minima
At the top of a smooth peak or the bottom of a valley the tangent is flat. Finding the greatest and least values with \(dy/dx = 0\), telling them apart with the second derivative, and the ends and corners where the rule is silent.
Maxima and minimaEnds and corners
6
Related Rates and Small Changes
When two changing quantities are tied by an equation, their rates are tied too. Differentiating a relation with respect to time, and estimating a small change with the tangent line.
Related rates of changeSmall changes and the tangent line
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- average rate of change
- The change in one quantity divided by the change in another over an interval, \(\Delta y/\Delta x\).
- chain rule
- The rule for a function of a function: if \(y\) depends on \(u\) and \(u\) on \(x\), then \(dy/dx = (dy/du)(du/dx)\).
- concave down
- Bending downward like a dome. The slope is decreasing and the second derivative is negative.
- concave up
- Bending upward like the inside of a cup. The slope is increasing and the second derivative is positive.
- continuous
- A graph with no break or jump, one that can be drawn without lifting the pen.
- derivative
- The rate of change of one quantity with respect to another at a single point, written \(dy/dx\). It is the limit of \(\Delta y/\Delta x\) as \(\Delta x\) tends to zero.
- implicit differentiation
- Differentiating an equation between \(x\) and \(y\) term by term, treating \(y\) as a function of \(x\), without first solving for \(y\).
- limit
- The value that an expression approaches as a variable is made closer and closer to some number, without ever being set equal to it.
- linear approximation
- The estimate \(f(x_0 + \Delta x) \approx f(x_0) + f'(x_0)\,\Delta x\), which follows the tangent instead of the curve for a small step.
- maximum
- A value of a quantity greater than all its values nearby. At a smooth maximum the tangent is flat.
- minimum
- A value of a quantity smaller than all its values nearby. At a smooth minimum the tangent is flat.
- normal
- The straight line through a point of a curve at right angles to the tangent there. If the tangent has slope \(m\), the normal has slope \(-1/m\).
- point of inflection
- A point where a graph changes from concave up to concave down, or back. The second derivative changes sign there.
- product rule
- The rule for the derivative of a product: \(d(uv)/dx = u\,dv/dx + v\,du/dx\).
- radian
- The unit of angle in which the arc of a circle equals the radius times the angle. One full turn is \(2\pi\) radians, and one radian is about \(57.3°\).
- second derivative
- The derivative of the derivative, \(d^2y/dx^2\). For a position against time it is the acceleration.
- stationary point
- A point where the derivative is zero and the tangent is flat. It may be a maximum, a minimum or neither.
- tangent
- The straight line that touches a smooth curve at one point and has the direction of the curve there. Close to the point, the curve and its tangent cannot be told apart.