Chapter 7: Differentiation

The chapter problem: the open box and the rising balloon

A sheet of card measures $16$ cm by $10$ cm. Four equal squares of side $x$ cm are cut from the corners, and the sides are folded up to make an open box.

The volume of the box is

The largest volume is not at the largest square, and it is not at the smallest. Calculus finds the exact value of $x$ that gives the greatest volume.

A spherical balloon is inflated for a display. Its volume increases at a steady rate of $100$ cm$^3$ per second, so

The radius $r$ grows too. Its rate of growth is linked to the rate at which the volume grows. Calculus finds $\dfrac{dr}{dt}$ at any radius, and shows how the rate changes as the balloon swells.

By the end of the chapter you will answer both parts exactly. You will write the volume of the box as a function of $x$, find where its gradient is zero, and prove that the point gives a maximum. You will also connect the two rates and find the speed at which the radius grows when $r = 5$ cm.

First contact — no working needed yet. Look at the box. If $x = 0$ the box is flat, and if $x = 5$ the card is cut in two and there is no box. Predict the value of $x$ between $0$ and $5$ that gives the greatest volume. Write your prediction down before you read on.

Then look at the balloon. It starts small and swells. Predict whether the radius grows faster at the start, or later when the balloon is large. Write one sentence to explain your prediction.

By the end of this chapter you will be able to

• estimate a gradient from a chord that shrinks, and explain the derivative as the gradient of the tangent;
• interpret a derivative as a rate of change, and read the notation $f'(x)$ and $\dfrac{dy}{dx}$;
• find a derivative from first principles for a whole-number power of $x$;
• differentiate $x^n$ for any rational $n$, with constant multiples, sums and differences;
• differentiate $e^{kx}$, $a^{kx}$, $\sin kx$, $\cos kx$, $\tan kx$ and $\ln x$;
• find and interpret a second derivative, and describe convex and concave sections;
• find stationary points and decide whether each is a maximum or a minimum;
• find the equation of a tangent and of a normal to a curve;
• decide where a function is increasing or decreasing;
• use the product, quotient and chain rules;
• solve connected-rates problems, and differentiate an inverse function;
• differentiate an implicit relation and a parametric curve;
• build a differential equation from a written description.

Where it is used

Differentiation is the mathematics of change. Engineers use it to design a container that uses the least material, economists use it to find the price that gives the greatest profit, and physicists use it for speed and acceleration. Every rate in a science is a derivative. The rules in this chapter turn a description of how a quantity changes into an equation, and that equation is where the next chapter, on integration, begins.

Your turn to connect it

Think of a quantity that grows or shrinks. It could be the water in a tank, the charge left in a battery, a savings balance, the height of a plant, or the temperature of a drink. Write down what you would measure, how fast it changes, and whether the rate itself changes as the quantity grows.

Roadmap

1. 7.1 The derivative as a gradient and a rate of change
2. 7.2 Differentiation from first principles
3. 7.3 Differentiating powers, exponentials, trigonometric functions and logarithms
4. 7.4 The second derivative, convexity and points of inflection
5. 7.5 Tangents, normals, maxima, minima and stationary points
6. 7.6 Increasing and decreasing functions
7. 7.7 The product, quotient and chain rules
8. 7.8 Connected rates of change and inverse functions
9. 7.9 Implicit and parametric differentiation
10. 7.10 Constructing differential equations
11. Review