Notes to a video lecture on http://www.unizor.com
Derivative Example -
Inverse Trigonometric Functions
f(x) = arcsin(x)
f I(x) = 1/√1−x²
We will use the method ofimplicit differentiation to obtain the formula for a derivative of this function.
Let's start from a definition of function arcsin(x).
The domain of this function is[−1,1] and its values are in[−π/2,π/2].
Then, for any value of the argument x from its domain the function value y is defined as an angle in radians such that
(b) −π/2 ≤ y ≤ π/2
The first statement can be expressed as
Since these two functions,A(x)=sin(arcsin(x)) andB(x)=x, are equal within domain [−1,1], their derivatives are equal as well.
The derivative of the A(x) can be obtained using the chain rule for compounded functions.
The derivative of B(x) is trivial.
BI(x) = d/dx[x] = 1
From equality of these two derivatives we conclude
Now let's analyze the expression cos(arcsin(x)).
We know that
cos(arcsin(x)) ≥ 0.
cos(arcsin(x)) = √1−x²
The final formula for a derivative is
[arcsin(x)]I = 1/√1−x²
A small detail remains when
Geometrically, it signifies that tangential lines at both ends,