Indeed, if
t = u(x) = x2 and v(t) = sin t, then
f (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2
Rationalised 2023-24
 MATHEMATICS
122
Put t = u(x) = x2 Observe that 
cos
dv
t
dt =
 and 
2
dt
dx =x
 exist Hence, by chain rule
df
dx  =
cos
2
dv
dt
t
x
dt
⋅dx
=
⋅
It is normal practice to express the final result only in terms of x Thus
df
dx  =
2
cos
2
2 cos
t
x
x
x
⋅
=
EXERCISE 5