How to take derivative of boundary conditions of non linear ODE

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I have a nonlinear boundary value problem to solve.

Boundary conditions are given as

r(x(t0),x(tf))=0

and I need to take the derivative of r with respect to x(to) and x(tf) respectively.

I am really confused about how to interpret r(x(t0),x(tf)) to go further with derivatives.

Any help would be appreciated .

Thanks