How do you differentiate #arcsin(sqrt(cos^2(1/x) )# using the chain rule?

1 Answer
Trevor Ryan.
Jan 7, 2016

#1/(sqrt(1-(cos^2(1/x)))) * 1/2(cos^2(1/x)^(-1/2)) * 2cos(1/x) * sin(1/x) * 1/x^2#.

Explanation:

#d/dx(sin^(-1)(sqrt(cos^2(1/x))))=1/(sqrt(1-(cos^2(1/x)))) * 1/2(cos^2(1/x)^(-1/2)) * 2cos(1/x) * sin(1/x) * 1/x^2#.

I have used the following rules in conjunction to reach my final answer :

#d/dx(sin^(-1)u(x))=1/sqrt((1-u^2))*(du)/dx#.

#d/dx[u(x)]^n=n[u(x)]^(n-1)*(du)/dx#.

#d/dxcosx=sinx#.

#d/dxx^n=nx^(n-1)#.

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