Question

.

Answer 1

@SithsAndGiggles

Answer 2

its just the equation x^n it doesnt say what n is

Answer 3

i typed it wrong one sec

Answer 4

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-2-examples-of-derivatives/MIT18_10SCF10_Ses2d.pdf

Answer 5

\[x^n+nx^{n-1} \Delta x+O((\Delta x)^2)\]

Answer 6

yes ime trying to find the derivative of x^n

Answer 7

\[\frac{d}{dx}[x^n]=\lim_{h\to0}\frac{(x+h)^n-x^n}{h}\]
(replace \(h\) with \(\Delta x\))
From the binomial theorem, you have
\[\begin{align*}(x+h)^n&=\sum_{k=0}^n\binom nkx^{n-k}h^k\\
&=\binom n0x^nh^0+\binom n1x^{n-1}h^1+\cdots+\binom n{n-1}x^{n-(n-1)}h^{n-1}+\binom nnx^{n-n}h^n\\
&=x^n+nx^{n-1}h+\cdots+nxh^{n-1}+h^n
\end{align*}\]
where \(\dbinom nk=\dfrac{n!}{k!(n-k)!}\).

Then you have
\[\frac{d}{dx}[x^n]=\lim_{h\to0}\frac{\left(x^n+nx^{n-1}h+\cdots+nxh^{n-1}+h^n\right)-x^n}{h}\]
The \(x^n\) terms disappear:
\[\frac{d}{dx}[x^n]=\lim_{h\to0}\frac{nx^{n-1}h+\cdots+nxh^{n-1}+h^n}{h}\]
and you can divide out a power of \(h\):
\[\frac{d}{dx}[x^n]=\lim_{h\to0}\left(nx^{n-1}+\frac{n(n-1)}{2}x^{n-2}h+\cdots+nxh^{n-2}+h^{n-1}\right)\]
All the terms with a factor of \(h\) appraoch 0, so you're left with the result of the power rule,
\[\frac{d}{dx}[x^n]=nx^{n-1}\]

Answer 8

ok gonna be like 10 mins for me to chew on this, then i might have a question but thanks.

Answer 9

yw

Answer 10

how come we can rewrite
\[(x+ \Delta x)^n = (x + \Delta x)^n(x+ \Delta x)^n ...(x+\Delta x)\]

as

\[x^n+n(\Delta x)x^{n-1}+O((\Delta x)^2)\]

i dont know where the x^n-1 comes in, or how you know the O((x)^2) is raised to the 2nd power there.

Answer 11

\((x+ \Delta x)^n = (x + \Delta x)(x+ \Delta x) ...(x+\Delta x)\) (n times)

The Binomial Theorem says:
\[ \large
(a + b)^{n} = \sum_{k = 0}^{n}\left(\begin{matrix}n \\ k\end{matrix}\right)a ^{n - k}b ^{k}
~~~~~~ \text{ where } \left(\begin{matrix}n \\ k\end{matrix}\right) = \frac{ n! }{ (n-k)! k! }
\]

Answer 12

this is a bit to confusing for me, where should i start...

Answer 13

Go through the link I messaged you earlier. It explains step-by-step the Binomial Theorem.