Chapter 6: Application of Derivatives

A derivative is more than a slope , it is a measure of instantaneous rate of change. In this chapter the derivative becomes a tool for solving practical problems: finding the steepest slope of a hill, the fastest growth rate of an investment, the dimensions of a box that uses the least material. We will translate each problem into a function $f(x)$, differentiate, and use the sign or vanishing of $f'$ to extract the desired information.

The chapter has three branches. The rate-of-change branch uses derivatives directly: if $y$ depends on $x$ and $x$ depends on time, then $\dfrac{dy}{dt} = \dfrac{dy}{dx}\dfrac{dx}{dt}$. The tangents/normals branch reads geometric information from $f'$. The optimisation branch , the heaviest in board examinations , finds extrema by setting $f'(x) = 0$ and classifying the critical points using $f''$.

For JEE, the same techniques become subtler. Problems often involve a parameter, asking for ranges where a function is increasing, or for the largest possible value of a complicated expression. The first-derivative test and the second-derivative test become indispensable. So does sign analysis: drawing a number line, marking the critical points, and checking the sign of $f'$ in each interval.

Approximation by differentials , using $f(a + h) \approx f(a) + h f'(a)$ , is a small but useful section. It allows quick estimates of square roots and trigonometric values without a calculator, and previews the deeper Taylor's theorem from higher mathematics.

The chapter has practical importance beyond examinations. Physics, economics, biology , every quantitative science uses optimisation. The derivative is the universal tool.

What's inside

1. Rate of change , derivatives as rates.
2. Increasing and decreasing functions , monotonicity test.
3. Tangents and normals , line equations from the derivative.
4. Approximation , using differentials.
5. Maxima and minima , first and second derivative tests.
6. Applied optimisation , modelling real problems.

Key results / Formula card

How to read this chapter

Master the first-derivative test, then the second. Practise translating word problems into functions of a single variable. The optimisation section repays repeated practice , most board questions on this chapter are optimisation problems in disguise.