Hi. Guys! Welcome to NumberX.
In this lecture; we are going to learn.
How to find derivative of sine inverse of x
using first principal or
definition of derivatives.
So, let us start--
Let sine inverse of x equals y.
Under given conditions
when x running from negative one to positive one,
and y
running from negative pi by 2 to pi by 2.
Under these conditions--
we can take sine on both sides of this equation
hence,
we get
x equals sine of y.
this equation is nothing but function of y.
Hence, we can write
f(y) = sin y
Instead of finding dy/dx;
we will find
dx/dy
ok.
So,
by definition of derivative;
we can write
f(y+h)
minus
f(y) upon h
when "h" tends to zero.
under limiting condition.
Now, we will simply put the value of
f(y+h) by replacing y
by (y+h) in this equation.
Hence, we get;
sin(y+h) - sin y
upon "h".
when "h" tends to zero.
ok.
Now, by using sin(A+B)
= sin A cos B + cos A sin B identity
we can write sin (y+h)
equals sin y cos h + cos y sin h
remaining terms will remain same.
Here, we will arrange the terms
and we can write this way. ok.
Advantage of writing in this form is
we can take common factor
sin y from these two terms.
and we get
sin y times (cos h - 1) upon h
plus
cos y times sin h upon h; when h tends to 0.
Since, y is not affected by variation of h;
hence, we can take it out
and we can write this way.
from here also
y is not dependent on h,
hence, we can take it out
and we can write this way.
Now,
we will
take -1 common from this expression
and we will get
negative sin y times limit
when h tends to zero of
(1 - cos h) upon h.
+ cos(y)
times limit of sin h upon h
when h tends to zero.
By using the limit formula,
Limit of (1-cos h) upon h,
when "h" tends to zero equals zero,
and limit of sin h upon h
when "h" tends to zero equals one
we can write
this expression as zero
and this equals one
Hence,
dx/dy = cos y
and reciprocal of this term
equals dy/dx
= 1 / cos y.
Now, our target is to write cosine y in terms of x
ok.
We know the identity
sin squared y plus cos squared y equals one
hence,
cos squared of y equals one minus sine squared of y.
This implies;
cos y = plus/minus
square root of the quantity  1 - sin squared of y
Since, we have taken sin y = x,
hence, we can replace sin of y by x
and we get
cosine of y
equals pm square root of the quantity one minus x squared.
But wait here;
Since, sine inverse of x,
lies in closed interval negatve pi/2 and pi/2
hence,
y also lies in this interval.
Because, we have taken y equals sine inverse of x.
This implies,
cosine y
lies in
closed interval [0,1]
because
y is an angle
which lie in 1st and 4th quadrant only.
ok.
But, one thing to note here,
since, cosine y is in the denominator of dy/dx
hence, it can't be equal to zero.
ok.
Now we will remove equality here and
cosine y
greater than zero and less than or equal to one.
By this inequality,
we can clearly say that-
cosine y is positive quantity.
Hence, we can remove
negative sign from here.
ok.
This implies,
dy/dx equals one over
the quantity square root of one minus x squared.
We can verify this fact by
graphing sin inverse of x on cartesion plane.
By end of this; we have finished this lecture.
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