
The chain rule
applications 
Derivative of
parametric functions, parametric derivatives 
Derivative of
parametric functions example

Derivative
of vectorvalued functions 





Derivative of
parametric functions, parametric derivatives 
When
Cartesian coordinates of a curve is represented as
functions of the same variable (usually written
t),
they are
called the parametric equations. 
Thus,
parametric equations in the xyplane 
x
= x (t)
and y
= y (t)
or x
= f (t)
and y
= g (t), 
denote
the x
and y
coordinate of the graph of a curve in the plane. 
Assume
that f
and g
are differentiable and f
'(t) is not
0 then, given parametric curve can be
expressed as y
= y (x) and this
function is differentiable at x,
that is 
x
= f (t)
or
t = f ^{1}(x), 
by plugging
into y
= g (t)
obtained is y
= g [f ^{1}(x)]. 
Therefore,
we use the chain rule and the derivative of the inverse function to find
the derivative of the parametric
functions, 



Derivative of
parametric functions example

Example:
Write
equation of the line tangent to the curve x
= t
+
1 and
y
=

t^{2}
+
4
at the point t
= 1.

Solution:
The equation x
=
t + 1
solve for t
and plug into y
=
 t^{2} + 4,
thus 
t
=
x
 1
=> y
=
 t^{2} + 4,
y
=
 (x
 1)^{2} + 4 
i.e., y

4
=
 (x
 1)^{2}
or y
=
 x^{2} + 2x + 3 translated parabola with the vertex V(x_{0},
y_{0}),
so V(1,
4). 
When
plotting points of a parametric curve by increasing t, the
graph of the function is traced
out in the direction of motion. 

The
derivative of the given parametric equations at t
= 1 is the slope of the tangent
line, 

since
t
=
x
 1
then m
=
y' (x)
=

2(x
 1)
= 
2(2
 1)
=

2,
m
=

2.

Therefore,
the equation of the line tangent to the given parametric curve at t
= 1 or the point P_{1}(2,
3) is 
y
 y_{1}
= m(x
 x_{1}),
y
 3
= 
2(x
 2) =>
y
= 
2x
+ 7. 

Derivative
of vectorvalued functions 
If
the radius vector r
of a point in a plane depends on a parameter t,
say t
represents time, so that its magnitude
and direction change continuously while t
changes, then its arrow sweeping out a curve. 
Let
r (t)
denotes its value at the moment t
and r (t
+ h)
represents its value at t
+ h,
and P
and P_{1}
are the corresponding
points of the curve, as is shown in the figure below. 
The
increment Dr
= r (t +
h)

r(t)
is the vector that falls in the direction of the secant line PP_{1
}
and points from P
to P_{1}.
The difference quotient 

obtained
by division with the scalar
h,
is the vector of the same
direction but of different length. 
The
limit of the difference quotient as
h
®
0 

is
the
derivative vector of the
vectorvalued
function that
falls in
the direction of the line tangent to the curve at P. 



If
x (t)
and y
(t)
are the scalar components of the vector r
(t)
then, according to rules of
vector algebra, 
Dr
= [x(t +
h)

x(t)] i
+ [y(t +
h)

y(t)] j 
and
by use of the definition of the derivative 

is
the derivative vector,
where x' (t)
and
y' (t)
are its scalar components, and where 

is
its length or magnitude. 









Calculus contents
C 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 