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## Main Question or Discussion Point

When deriving the Runge-Kutta Method to solve y'=f(x) we need to use Taylor expansion. Hence we need to differentiate the function many times.

y'(x)=f(y(x))

y''(x)=f'(y(x))y'(x) = f'(y(x))f(y(x))

y''' = f''(y(x))(f(y(x)),y'(x)) + f'(y(x))f'(y(x))y'(x)

I can understand the second derivative but not the third derivative especially the term

f''(y(x))(f(y(x)),y'(x))

Can someone please explain the meaning of the notation used here.

y'(x)=f(y(x))

y''(x)=f'(y(x))y'(x) = f'(y(x))f(y(x))

y''' = f''(y(x))(f(y(x)),y'(x)) + f'(y(x))f'(y(x))y'(x)

I can understand the second derivative but not the third derivative especially the term

f''(y(x))(f(y(x)),y'(x))

Can someone please explain the meaning of the notation used here.