Class 12 Maths Application of Derivatives Notes

6.1 Introduction

The derivative of a function gives the rate at which a quantity changes. Beyond slopes of curves, derivatives are used to:

This chapter focuses on applications of derivatives in various scenarios.

The derivative $\frac{dy}{dx}$ dxdy gives the instantaneous rate of change of $y$ y with respect to $x$ x.
Examples:
- Velocity is the rate of change of displacement: $v = \frac{dx}{dt}$ v=dtdx
- Marginal cost is the rate of change of total cost with respect to quantity: $MC = \frac{dC}{dq}$ MC=dqdC

Derivatives are a mathematical tool to measure change.

Increasing Function: $f'(x) > 0$ f′(x)>0 → the function rises as x increases
Decreasing Function: $f'(x) < 0$ f′(x)<0 → the function falls as x increases

Critical points occur where $f'(x) = 0$ f′(x)=0. These points are used to determine maxima, minima, or points of inflection.

Find the derivative $f'(x)$ f′(x)
Solve $f'(x) = 0$ f′(x)=0 → critical points
Use second derivative test:
- $f”(x) > 0$ f′′(x)>0 → Local Minimum
- $f”(x) < 0$ f′′(x)<0 → Local Maximum
Alternatively, use the first derivative test by checking sign changes of $f'(x)$ f′(x)