Q&A

# Differentiating a series expansion of an arbitrary function

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Differentiate $F(x)=f(x)+(a+h-x)f'(x)+\frac{(a+h-x)^2}{2!}f''(x)+... + \frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n-1)}(x)+k(a+h-x)^m$

I was trying to solve it following way.

$$F'(x)=f'(x)-f'(x)+(a+h-x)f''(x)-(a+h-x)f''(x)+\color{blue}{\frac{(a+h-x)^2}{2!}f'''(x)}+\frac{\color{red}{(n-1)}(a+h-x)^{n-2}}{(n-1)!} f^{(n-1)}x+\frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n)}(x)+km(a+h-x)^{m-1}(-1)$$

$$= \frac{(a+h-x)^2}{2!}f'''(x)+\frac{(n-1)(a+h-x)^{n-2}}{(n-1)!}f^{(n-1)}(x)+\frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n)}(x)-km(a+h-x)^{m-1}$$

I had marked(with Red color) something. According to my book, it shouldn't be there. But, according to my calculation it should be there.

They didn't write the blue line also but, I don't have any problem on it cause, they put that as extra function (used ... for that function).

The function I said to differentiate. The way they differentiate. If it canceled with $(n-1)!$ than, why it didn't disappear from both side (numerator and denominator)? Even, why the value decreased if it canceled?

In last line $$\frac{(a+h-x)^{n-2}}{(n-2)!} f^{(n-1)}x$$ where it had gone? I can merge the denominator to $\frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n)}(x)$ but, how to merge numerator?

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$$\frac{n-1}{(n-1)!}$$ $$=\frac{n-1}{(n-1)(n-2)!}$$ $$=\frac{1}{(n-2)!}$$

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