Application of Derivatives: Class 12 Maths · Chapter 6

6.1 Introduction

A derivative is more than a symbol to compute. It answers practical questions. How fast is a puddle spreading right now? Is a company's profit still climbing or has it begun to fall? What size of box holds the most from one sheet of card? In Chapter 5 you learnt to find derivatives of many kinds of function. In this chapter you will use them.

We follow three threads (Fig. 6.1). First, a derivative measures a rate of change, and the chain rule lets one known rate give us another. Second, the sign of $f^{\mathrm{\prime }}(x)$ tells us where a graph goes up and where it goes down. Third, the places where $f^{\mathrm{\prime }}(x)$ is zero or does not exist are the only interior places where a function can reach a peak or a dip. That gives us a method for finding the largest or smallest value a quantity can take.

6.2 Rate of Change of Quantities

Suppose a quantity $y$ depends on another quantity $x$ through a rule $y=f(x)$. The derivative $\frac{dy}{dx}$, also written $f^{\mathrm{\prime }}(x)$, tells us how many units $y$ changes for each unit change in $x$ at that instant. Evaluated at one particular input, ${\frac{dy}{dx}\mid }_{x=x_0}$ or $f^{\mathrm{\prime }}(x_0)$ measures how fast $y$ is changing at the moment $x$ passes $x_0$.

Speed is the familiar case. If $s$ metres is the distance a train has covered after $t$ seconds, then $\frac{ds}{dt}$ is its speed in metres per second. The units of a rate are always "units of the top quantity per unit of the bottom quantity". Writing them down catches many mistakes.

Often two quantities $x$ and $y$ both change with time $t$, say $x=f(t)$ and $y=g(t)$. If we know how fast each changes with time, the chain rule gives the rate of $y$ with respect to $x$:

Read the other way round, $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$. This is the workhorse of related rates problems: a formula connects two quantities, one rate is known, and we want the other.

Here is a picture that explains a rate we will use again and again. When the radius of a circle grows by a small amount $\mathrm{\Delta }r$, the new area is a thin ring around the old circle (Fig. 6.2). Cut the ring and straighten it: it is almost a strip of length $2\pi r$ and width $\mathrm{\Delta }r$. So $\mathrm{\Delta }A\approx 2\pi r\mathrm{\Delta }r$, and dividing by $\mathrm{\Delta }r$ gives $\frac{dA}{dr}=2\pi r$, which is exactly the circumference.

A note on signs. If $y$ grows as $x$ grows, $\frac{dy}{dx}$ is positive; if $y$ shrinks as $x$ grows, it is negative. So a quantity that is falling has a negative rate. In Example 2 the answer $-20\mathrm{c}{\mathrm{m}}^2\mathrm{/}\mathrm{m}\mathrm{i}\mathrm{n}$ is not an error: the minus sign says "decreasing".

Rates matter in business too. If $C(x)$ is the total cost of producing $x$ units, the marginal cost is $C^{\mathrm{\prime }}(x)$: the rate at which cost grows with output at that level of production. It is close to the extra cost of making one more unit. In the same way, if $R(x)$ is the total revenue from selling $x$ units, the marginal revenue is $R^{\mathrm{\prime }}(x)$.

6.3 Increasing and Decreasing Functions

Picture the temperature in Jaipur through a summer day. From dawn it climbs, it peaks in the afternoon, and then it falls through the evening. Over the morning hours a later time always has a higher temperature; over the evening a later time always has a lower one. Those two behaviours, increasing and decreasing, are what we now define and then test with derivatives.

Take two inputs $x_1<x_2$ from an interval and compare the outputs (Fig. 6.3). On the rising part of a graph the later input gives the larger output. On the falling part it gives the smaller one. If the two inputs sit on opposite sides of a peak, either can win, so the function is neither increasing nor decreasing on an interval that contains the peak.

A function that is increasing throughout an interval, or decreasing throughout it, is called monotonic on that interval. We also speak of behaviour at a point: $f$ is increasing at $x_0$ if there is some open interval containing $x_0$ on which $f$ is increasing, and similarly for decreasing. This is a local idea. A function can be increasing at a point and yet be decreasing elsewhere in its domain.

Checking every pair of points is fine for straight lines, but hopeless for most functions. The derivative gives a far quicker test. At a point where the graph rises, the tangent slopes upward, so $f^{\mathrm{\prime }}(x)>0$; where it falls, the tangent slopes downward and $f^{\mathrm{\prime }}(x)<0$ (Fig. 6.4). The theorem below turns this picture into a proof.

A useful extension. The conclusion survives a few awkward points. If $f$ is continuous on an interval and $f^{\mathrm{\prime }}(x)>0$ at every point except the end points, $f$ is strictly increasing on the whole interval, end points included. The same holds if $f^{\mathrm{\prime }}$ is zero only at a few isolated points: $f(x)=x^3$ has $f^{\mathrm{\prime }}(0)=0$ and is still strictly increasing on $𝐑$.

To find where a function increases or decreases, we use the points at which $f^{\mathrm{\prime }}(x)=0$ (or $f^{\mathrm{\prime }}$ fails to exist) as fence posts. They cut the real line into intervals. Inside each interval $f^{\mathrm{\prime }}$ cannot change sign if it is continuous, so testing one point per interval is enough.