What is the derivative of this function y=sin^-1(1/x)?
1 Answer
Jul 13, 2016
Explanation:
color(orange)"Reminder" d/dx(sin^-1x)=1/(sqrt(1-x^2) here, however, x =
1/x Differentiate using the
color(blue)" chain rule combined with power rule"
color(orange)" Chain rule"
color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|)))........ (A)
"---------------------------------------------------------------"
f(g(x))=sin^-1(1/x)rArrf'(g(x))=1/(sqrt(1-(1/x)^2) and
g(x)=1/x=x^-1rArrg'(x)=-x^-2=-1/x^2
"-----------------------------------------------------------------"
Substitute these values into (A)
=1/(sqrt(1-1/x^2))xx-1/x^2
=(-1)/(x^2sqrt(1/x^2(x^2-1))
=(-1)/(x^2xx1/xsqrt(x^2-1)
rArrd/dx(sin^-1(1/x))=(-1)/(xsqrt(x^2-1)
