Find derivative by first principle method

Question

How to find the derivative of cos^-1 x / x by first principle method ?

Answer 1

in computing function's derivative, will need to know d(cos^-1x)/dx

for this piece, cos(cos^-1x)=x

d(cos(cos^-1x))/dx=1

-sin(cos^-1x)*d(cos^-1x)dx=1 ,,,,,chain rule for composite function

sin(cos^-1x) = (+/-)√(1-x²)

the principle branch for cos^-1x is from 0≤cos^-1x<π

and the slope of the tangent to the curve is negative in this branch

note: slope of curve is change in cos^-1x divided by change in x

where positive change of x is in direction of -1<x≤1

so d(cos^-1x)/dx=(-)1/√(1-x²)

for d((cos^-1x)/x)/dx use product rule

d((cos^-1x)/x)/dx=-[(cos^-1x)/x²+(x/x²)*d(cos^-1x)/dx

=(-1/x²)[(cos^-1x) +x/√(1-x²)]