Application of Derivatives — Study Notes

1. Introduction

The derivative measures the rate at which a quantity changes, and this chapter turns that single idea into a wide array of applications. From finding the slope of a tangent and writing equations of tangents and normals, to determining where functions are increasing or decreasing, to locating maxima and minima, the derivative becomes the primary tool for analysing the behaviour of functions.

The chapter begins with the rate of change of a quantity, the geometrical interpretation of the derivative, and the equations of tangents and normals. It then develops the tests for monotonicity using the sign of the derivative, and culminates in the study of turning points through the first derivative test and the second derivative test. Applications to real-world optimisation problems, such as maximising area or minimising cost, complete the chapter.

Beyond the board examination, the ideas here are the foundation of optimisation in economics, physics, and engineering. The concept of marginal cost and marginal revenue in economics is exactly a derivative, and the design of structures, containers, and motion paths all involve maximising and minimising. A clear understanding of when a function increases or decreases, and how to classify critical points, is among the most valuable skills in applied mathematics.

2. Rate of Change of Quantities

If a quantity y changes with x according to y = f(x), then the rate of change of y with respect to x at x = x0 is f'(x0). In many applied problems, x itself depends on time t, and by the chain rule:

$$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$$

This is the basis of related-rates problems, where two changing quantities are connected by an equation and we relate their rates of change. For example, if a balloon is being inflated, the rate of change of the radius can be related to the rate of change of the volume through $V = \frac{4}{3}\pi r^3$.

3. Tangents and Normals

The slope of the tangent to the curve y = f(x) at the point (x0, y0) is f'(x0).

Equation of the Tangent

$$y - y_0 = f'(x_0)(x - x_0)$$

Equation of the Normal

Since the normal is perpendicular to the tangent, its slope is the negative reciprocal $-\frac{1}{f'(x_0)}$, provided f'(x0) $\neq$ 0:

$$y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$$

Special Cases

• If the tangent is parallel to the x-axis, f'(x0) = 0 and the tangent is horizontal.
• If the tangent is parallel to the y-axis, the curve has a vertical tangent, where the derivative tends to infinity.

4. Increasing and Decreasing Functions

Let f be a function defined on an interval I.

• f is strictly increasing on I if for any x1 < x2 in I, f(x1) < f(x2).
• f is strictly decreasing on I if for any x1 < x2 in I, f(x1) > f(x2).

Test Using the Derivative

• If f'(x) > 0 for all x in I, then f is strictly increasing on I.
• If f'(x) < 0 for all x in I, then f is strictly decreasing on I.
• If f'(x) = 0 for all x in I, then f is constant on I.

To find the intervals of increase and decrease, solve f'(x) = 0 to get the critical points, divide the real line into intervals, and test the sign of f' in each interval.

5. Approximations Using Differentials

For small changes, the differential dx approximates the change in x, and

$$dy = f'(x)\, dx$$

The actual change $\Delta y = f(x + \Delta x) - f(x)$ is approximated by $dy$ when $\Delta x$ is small. The approximate value is then

$$f(x + \Delta x) \approx f(x) + f'(x) \Delta x$$

For example, to approximate $\sqrt{36.1}$, take $f(x) = \sqrt{x}$, x = 36, dx = 0.1:

$$\sqrt{36.1} \approx 6 + \frac{1}{2\sqrt{36}}(0.1) = 6 + \frac{0.1}{12} = 6.0083$$

6. Maxima and Minima

A function f has a local maximum at x = c if f(c) is greater than or equal to f(x) for all x near c, and a local minimum if f(c) is less than or equal to f(x) for all x near c. The points where f'(c) = 0 or f'(c) does not exist are called critical points.

First Derivative Test

At a critical point c where f'(c) = 0: - If f' changes sign from positive to negative at c, then c is a point of local maximum. - If f' changes sign from negative to positive at c, then c is a point of local minimum. - If f' does not change sign, c is a point of inflexion.

Second Derivative Test

If f'(c) = 0 and f''(c) exists: - If f''(c) < 0, then c is a point of local maximum. - If f''(c) > 0, then c is a point of local minimum. - If f''(c) = 0, the test is inconclusive and the first derivative test must be used.

Absolute (Global) Extrema on a Closed Interval

The absolute maximum and minimum of a continuous function on [a, b] occur either at critical points in (a, b) or at the endpoints. Evaluate f at all critical points and at a and b, and compare.

7. Applications in Optimisation Problems

Real-world optimisation problems follow a standard procedure: 1. Identify the quantity to be maximised or minimised and write it as a function of one variable. 2. Use the given constraints to eliminate the other variables. 3. Find the critical points by setting the derivative to zero. 4. Use the second derivative test to confirm the nature of each critical point. 5. Answer the question, considering practical domain restrictions.

For example, to find two positive numbers whose sum is 24 and whose product is as large as possible: if one number is x, the other is 24 - x, so P = x(24 - x) = 24x - x^2. Then P' = 24 - 2x = 0 gives x = 12, and P'' = -2 < 0 confirms a maximum. Both numbers are 12, with product 144.

Quick Revision Tables

Table 1: Equation of Tangent and Normal

Table 2: Monotonicity and Extrema Tests

Table 3: Differential Approximation

Mind Map

Important Diagrams (SVG)

Diagram 1: Tangent and Normal to a Curve

Diagram 2: Local Maximum, Local Minimum and Point of Inflexion

Common Mistakes

1. Writing the normal's slope as the same as the tangent's slope. The normal is perpendicular, so its slope is the negative reciprocal.
2. Forgetting to divide by the derivative correctly in related-rates problems; always relate dy/dt and dx/dt through the chain rule before substituting numbers.
3. Declaring a function increasing on an interval from a single sample point. The sign of f' must be verified throughout the whole interval.
4. Concluding a point is a maximum purely from f'(c) = 0; the sign change of f' must be checked.
5. Applying the second derivative test when f''(c) = 0; it is inconclusive and the first derivative test must be used.
6. In optimisation problems, forgetting the domain restrictions of the variable (for example, lengths must be positive), which may eliminate some candidates.
7. Using the first derivative test at a point where f' does not exist without separately considering differentiability.
8. For absolute extrema on [a, b], forgetting to evaluate f at the endpoints a and b.
9. Confusing dy (the differential) with delta y (the actual change); they are equal only in the limit.
10. Solving only f'(x) = 0 but ignoring points where the derivative is undefined when listing critical points.

Exam Tips

1. In tangent/normal questions, always first compute the slope m = f'(x0), then write the tangent and normal equations; verify the answer passes through (x0, y0).
2. For monotonicity, solve f'(x) = 0, draw a number line, and test one point in each interval; present the intervals with correct notation.
3. For optimisation word problems, define variables clearly, form the function, note the domain, and use the second derivative to justify maxima/minima.
4. In related-rates questions, differentiate the connecting equation before substituting the numerical values, to avoid plugging in constants prematurely.
5. For the second derivative test, if f''(c) = 0, switch to the first derivative test and describe the sign change.
6. To check whether a function is strictly increasing, remember that f'(x) >= 0 with f' not identically zero on the interval is usually enough for continuous functions.
7. Practise the standard optimisation problems: greatest product with fixed sum, largest area of a rectangle with fixed perimeter, and least surface area for a given volume of a cylinder.
8. For approximation questions, choose x as a perfect value close to the target and dx as the small difference; keep computations to a few decimals.

Conclusion

The application of derivatives converts the abstract notion of the derivative into a practical tool of enormous reach. Tangent and normal equations connect calculus to coordinate geometry, the sign of the derivative organises the graph of a function into increasing and decreasing regions, and the first and second derivative tests locate and classify the peaks and troughs of a function. Differential approximation provides quick numerical estimates, while the optimisation problems tie everything to real-world decision making. Every idea in this chapter is used repeatedly in the chapters that follow, especially in curve sketching, in the geometric interpretation of integrals, and in the analysis of solutions of differential equations. For the board examination, this is a high-scoring chapter where methodical presentation of each step is rewarded, and for the future, it is the gateway to genuine mathematical modelling.