Solve the equation x-sinx=0.

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In order to solve the equation, which is a transcedental one, you have to find the derivative of the function f.

Yes, we can find the derivative of the function f, because is a continuous function, formed by elementary functions as the linear one,x , and the trigonometrical one, sin x!

f'(x)=1-cosx

We can note that is a monotone increasing function

( -1<cosx<1), so the difference 1 -cos x>0 =>f(x)>0, so f(x) is an injection.

We can also do a very simple calculus:

f(0)=0-sin0=0-0=0.

We've demonstrated earlier before that f(x) is an injection, so, x=0 is the only solution for x-sinx=0

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