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I'm having trouble getting my head around the expansion of: $$f(y+bhf(y))$$ I want to expand a function within a function. I have been given that $$f(y+hf(y))=f(y)+hf(y)f'(y)+\frac{h^2f(y)^2}{2}f''(y)+\cdots$$ I assume, therefore that my original expansion would follow from this as $$f(y+bhf(y))=f(y)+bhf(y)f'(y)+\frac{b^2h^2f(y)^2}{2}f''(y)+\cdots$$ from which I can see the basic Taylor series form, but I don't quite understand where each bit has gone. How do the multiple parts expand out?

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I worked it out. Using the standard Taylor Series: $$f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots$$ with $x=y+bhf(y)$ and $a=y$, the solution pops out.

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