The derivative of
arcsin(x) is
1−x21.
Solution. Let
y=sin−1(x). Then
x=sin(y) and
−2π≤y≤2π. We will differentiate with respect to
x:
dxdx=dxdsin(y)⟺1=dyd(sin(y))dxdy⟺1=cos(y)dxdy⟺dxdy=cos(y)1,
where we have seen
here that
dyd(sin(y))=cos(y). We have that
cos(y)≥0 if
−2π≤y≤2π. Since
sin2(y)+cos2(y)=1, we have that:
cos2(y)=1−sin2(y)⟺cos(y)=1−sin2(y).
where
sin2(y)=x2 since
sin(y)=x. Substituting everything, we get that the derivative of
arcsin(x) is:
dxdarcsin(x)=dxdsin−1(x)=dxdy=cos(y)1=1−x21,x∈(−1,1).
So, the derivative of
arcsin(x) is
1−x21.