
Differentiation 4-41
Thus,
g' x
d
dx
x
x
x
d
dx
x x
d
dx
x
( )
cosh
( ) (cosh ) cosh ( )
(
=
+
=
+ − +
1
1 1
11
1
1
2
2
+
=
+ −
+
x
x x x
x
)
( )sinh cosh
( )
.
(by quotient rule)
4.10 DIFFERENTIATION OF PARAMETRIC EQUATIONS
Sometimes both x and y are expressed in terms of a third variable, say t. This independent variable
is called parameter, and the equations x = f
1
(t) and y = f
2
(t), thus given are known as parametric
equations. In the case of parametric equations, we find dy/dx in the following manner:
dy
dx
dy
dt
dt
dx
dy dt
dx dt
= ⋅ =
/
/
.
Example 4.38 Find
dy
dx
if x = t - 2t
2
and y = 2t - t
3
.
Solution: We have
dy
dt
t
dx
dt
t
= −
= −
2 3
1 4
2
∴ =
−
−
dy
dx
dy dt
dx dt
t
t
=