Introduction

In this chapter, we will explore various applications of derivatives, which play a fundamental role in understanding the behavior of functions. Some key applications include:

• Rate of Change: The derivative represents the instantaneous rate of change of a function with respect to its variable.

• Approximating Function Values: Differentials help estimate function values near a given point, providing useful approximations when exact calculations are difficult.

• Monotonicity Behavior: The derivative determines where a function is increasing or decreasing. If \( f'(x) > 0 \), the function is increasing, and if \( f'(x) < 0 \), the function is decreasing.

• Finding Maximum and Minimum Values: Extrema occur at critical points where \( f'(x) = 0 \) or is undefined. The first and second derivative tests help classify these points as local maxima or minima.

• Tangents and Normals: The derivative gives the slope of the tangent to a curve at a given point. The normal is perpendicular to the tangent, and their equations help in geometric and physical interpretations of curves.

• Other Applications: These include identifying points of inflection, analyzing concavity, curvature etc