® Chain rule is rule to differentiate composites of functions If f = v o u, t = u (x)
and if both dt
dx  and dv
dt  exist then
df
dv dt
dx
=dt dx
⋅
® Following are some of the standard derivatives (in appropriate domains):
(
)
1
2
1
sin
1
d
x
dx
x
−
=
−
(
)
1
2
1
cos
1
d
x
dx
x
−
= −
−
(
)
1
2
1
tan
1
d
x
dx
x
−
=
+
(
x)
x
d
e
e
dx
=
(
)
1
log
d
x
dx
x
=
® Logarithmic differentiation is a powerful technique to differentiate functions
of the form f (x) = [u (x)]v (x) Here both f (x) and u(x) need to be positive for
this technique to make sense —v
v
v
v
v—
Rationalised 2023-24
v With the Calculus as a key, Mathematics can be successfully applied
to the explanation of the course of Nature