NCERT Book Class 12 Maths Chapter 6 Applications of Derivatives

Chapter 6
APPLICATION OF
DERIVATIVES
v With the Calculus as a key, Mathematics can be successfully applied
to the explanation of the course of Nature.” — WHITEHEAD v

6.1 Introduction
In Chapter 5, we have learnt how to find derivative of composite functions, inverse
trigonometric functions, implicit functions, exponential functions and logarithmic functions.
In this chapter, we will study applications of the derivative in various disciplines, e.g., in
engineering, science, social science, and many other fields. For instance, we will learn
how the derivative can be used (i) to determine rate of change of quantities, (ii) to find
the equations of tangent and normal to a curve at a point, (iii) to find turning points on
the graph of a function which in turn will help us to locate points at which largest or
smallest value (locally) of a function occurs. We will also use derivative to find intervals
on which a function is increasing or decreasing. Finally, we use the derivative to find
approximate value of certain quantities.
6.2 Rate of Change of Quantities
ds
Recall that by the derivative , we mean the rate of change of distance s with
dt
respect to the time t. In a similar fashion, whenever one quantity y varies with another
dy
quantity x, satisfying some rule y = f ( x ) , then (or f ′(x)) represents the rate of
dx
dy 
change of y with respect to x and dx  (or f ′(x0)) represents the rate of change
x = x0

of y with respect to x at x = x0 .
Further, if two variables x and y are varying with respect to another variable t, i.e.,
if x = f (t ) and y = g (t ) , then by Chain Rule
dy dy dx dx
= , if ≠0
dx dt dt dt