

Applications of the derivative 
Tangent,
normal, subtangent and subnormal 
Property of the parabola 
Property
of power functions 
Property
of the exponential function 






Tangent,
normal, subtangent and subnormal 
A
segment of a tangent to a curve lying between the tangency point (the
point at which a tangent is drawn to a
curve) and the intercept of the tangent with the xaxis
is called the length of the tangent. 
A
segment of the xaxis
lying between the xcoordinate
of the tangency point and the intercept of the tangent with
the axis is called the subtangent. 
Therefore,
the subtangent is the projection of the segment of the tangent onto the xaxis. 
A
segment of a line normal to a tangent lying between the tangency point
and the intercept of the normal with the xaxis
is called the length of the normal. 
A
segment of the xaxis
lying between the xcoordinate
of the tangency point and the intercept of the normal with
the axis is called the subnormal. 
Therefore,
the subnormal is the projection of the segment of the normal onto the xaxis. 
In
the figure below denoted are, 
the
length of the tangent t_{l}
= PT,
the subtangent s_{t}
= TX, 
the
length of the normal n_{l}
= PN
and the subnormal s_{n}
= NX. 

In
the right triangle PTX, 



In
the right triangle PNX, 


Therefore,
the subtangent, 
and
the subnormal 


The
length of the tangent, 
and
the
length of the normal 



Properties
of the parabola 
At
every point of the parabola y^{2}
= 2px the subnormal have the
same value p. 
By
differentiating both sides of the equation of the 
parabola with
respect to x
we get 
2yy'
= 2p
or yy'
= p = s_{n}. 
We use this property to construct the normal and
the 
tangent at a point of the parabola. 
Thus,
the normal line passes through the given point 
P(x,
y) of the
parabola
and the point x
+ p,
lying on the 
xaxis. 
Further,
since 





therefore,
the
vertex of the parabola y^{2}
= 2p bisects the
subtangent. 

Property
of power functions 
The
subtangent of the power function y
= ax^{n},
where n
is a positive integer, 


Thus
for example, the tangent to the parabola y
= ax^{2}
at P (x,
y)
bisects the abscissa of P
that is,
s_{t}
= x/2,
as
shows the figure below. 


Property
of the exponential function 
The
subtangent of the exponential function
y
= e^{rx},
r
Î
R and r ¹
0, since 


Thus
for example, at every point of the exponential function
y
= e^{x} the
subtangent s_{t}
= 1,
as shows the figure
above. 








Functions
contents F 



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