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#4: Post edited
by
Mithrandir24601
·
2021-07-11T14:50:47Z (over 3 years ago)
Edited title so Latex doesn't overflow
Differentiate $F(x)=f(x)+(a+h-x)f'(x)+\frac{(a+h-x)^2}{2!}f''(x)+... + \frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n-1)}(x)+k(a+h-x)^m$
- Differentiating a series expansion of an arbitrary function
#3: Post edited
- >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).
- <hr/>
- The function I said to differentiate.
- [![enter image description here][1]][1]
- The way they differentiate.
- [![enter image description here][2]][2]
- <hr/>
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?- [1]: https://i.stack.imgur.com/Z9mLK.png
- [2]: https://i.stack.imgur.com/x1yGY.png
- >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).
- <hr/>
- The function I said to differentiate.
- [![enter image description here][1]][1]
- The way they differentiate.
- [![enter image description here][2]][2]
- <hr/>
- <s>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?</s>
- 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?
- [1]: https://i.stack.imgur.com/Z9mLK.png
- [2]: https://i.stack.imgur.com/x1yGY.png
#2: Post edited
- >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).
- <hr/>
- The function I said to differentiate.
- [![enter image description here][1]][1]
- The way they differentiate.
- [![enter image description here][2]][2]
- [1]: https://i.stack.imgur.com/Z9mLK.png
- [2]: https://i.stack.imgur.com/x1yGY.png
- >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).
- <hr/>
- The function I said to differentiate.
- [![enter image description here][1]][1]
- The way they differentiate.
- [![enter image description here][2]][2]
- <hr/>
- 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?
- [1]: https://i.stack.imgur.com/Z9mLK.png
- [2]: https://i.stack.imgur.com/x1yGY.png
#1: Initial revision
Differentiate $F(x)=f(x)+(a+h-x)f'(x)+\frac{(a+h-x)^2}{2!}f''(x)+... + \frac{(a+h-x)^{n-1}}{(n-1)!}f^{(n-1)}(x)+k(a+h-x)^m$
>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).
<hr/>
The function I said to differentiate.
[![enter image description here][1]][1]
The way they differentiate.
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/Z9mLK.png
[2]: https://i.stack.imgur.com/x1yGY.png