2016-03-01-thm-derivatives-polynomials.md

title	lang	category	permalink	ident	parent	kind	mathjax	layout	type
Derivatives of Polynomials	en	en	en/theorem_derivatives_polynomials	theorem_derivatives_polynomials	derivatives	theorem	true	post	post

$(c)'=0$

Proof

$f(x)=c$;

$\displaystyle\lim_{h\longrightarrow 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\longrightarrow 0}\dfrac{c-c}{h}=\lim_{h\longrightarrow 0} 0 = 0$

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$(x)'=1$

Proof

$f(x)=x$;

$\displaystyle\lim_{h\longrightarrow 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\longrightarrow 0}\dfrac{(x+h)-x}{h}=\lim_{h\longrightarrow 0} 1 = 1$

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$(x^2)'=2x$

Proof

$f(x)=x^2$;

$\displaystyle\lim_{h\longrightarrow 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\longrightarrow 0}\dfrac{(x+h)^2-x^2}{h}=\\\displaystyle\lim_{h\longrightarrow 0} \dfrac{x^2+2xh+h^2-x^2}{h}=\lim_{h\longrightarrow 0} 2x+h = 2x$

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$(x^n)'=nx^{n-1}$

Proof

$f(x)=x^n$;

According to the binomial expansion {% cite theorem_binomial_expansion %} <br><br> 

$(x+h)^n=x^n+nx^{n-1}h+\binom{n}{2}x^{n-1}h^2+\cdots+\binom{n}{n-2}x^2h^{n-2}+nxh^{n-1}+h^n$ <br><br> 

So <br><br> 

$\displaystyle\lim_{h\longrightarrow 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\longrightarrow 0}\dfrac{(x+h)^2-x^2}{h}=\\
\displaystyle\lim_{h\longrightarrow 0} \dfrac{x^n+nx^{n-1}h+\binom{n}{2}x^{n-1}h^2+\cdots+\binom{n}{n-2}x^2h^{n-2}+nxh^{n-1}+h^n-x^n}{h}=\\
\displaystyle\lim_{h\longrightarrow 0} nx^{n-1}+\binom{n}{2}x^{n-1}h+\cdots+\binom{n}{n-2}x^2h^{n-3}+nxh^{n-1}+h^{n-1} =\\
 nx^{n-1}$