In a similar fashion, whenever one quantity y varies with another
quantity x, satisfying some rule 
( )
y
=f x
, then dy
dx  (or f′(x)) represents the rate of
change of y with respect to x and  
dy
dx
x x

= 0
 (or f′(x0)) represents the  rate of change
of y with respect to x at 
0
x
=x if Further, if two variables x and y are varying with respect to another variable t, i e ,
( )
x
=f t
and 
( )
y
=g t
, then by Chain Rule
dy
dx  = dy
dx
dt
dt ,  if  
0
dx
dt ≠
Chapter 6
APPLICATION OF
DERIVATIVES
Rationalised 2023-24
 MATHEMATICS
148
Thus, the rate of change of y with respect to x can be calculated using the rate of
change of y and that of x both with respect to t