Hipi Zhdripi i Matematikës/1270

3.2. TEOREMAT PËR DERIVATET        T e o r e m a  3.2.1. - Derivati i konstantes është i barabartë me zero, pra: $(c{)}^{\prime }=0,c=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}$ (38)        V ë r t e t i m Meqenëse këtu $(\mathrm{\forall }x\in X)f(x)=c$, prandaj $f(x+\mathrm{\Delta }x)=c$, $\mathrm{\Delta }y=f(x+\mathrm{\Delta }x)-f(x)=0$, ndërkaq: $f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{0}{\mathrm{\Delta }x}=0$, çka do të thotë se $(c{)}^{\prime }=0$.        T e o r e m a  3.2.2 - Derivati i argumentit është i barabartë me $1$, pra: $(x{)}^{\prime }=1$. (39)        V ë r t e t i m Këtu $f(x)=x$, $f(x+\mathrm{\Delta }x)=x+\mathrm{\Delta }x$, dhe $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}=\frac{x+\mathrm{\Delta }x-x}{\mathrm{\Delta }x}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }x}=1$ ndërsa vlera kufitare e këtij raporti, kur $\mathrm{\Delta }x\to 0$, është: $f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}1=1$.         Pra, konkludojmë: $(x{)}^{\prime }=1$.        T e o r e m a  3.2.3. - Derivati i shumës algjebriketë dy e më tepër funksioneve është i barabartë me shumën algjebrike të derivateve të tyre, pra: $(u+v-w{)}^{\prime }=u^{\prime }+v^{\prime }-w^{\prime }$. (40)        V ë r t e t i m Le të marrim funksionin $f(x)$ që është shprehur si shumë algjebrike e tri funksioneve : $f(x)=u(x)+v(x)-w(x)$. Shtesa e këtij funksioni është: $\begin{array}{cc}\mathrm{\Delta }y& =[u(x+\mathrm{\Delta }x)+v(x+\mathrm{\Delta }x)-w(x+\mathrm{\Delta }x)]-[u(x)+v(x)-w(x)]\\ & =[u(x+\mathrm{\Delta }x)-u(x)]+[v(x+\mathrm{\Delta }x)-v(x)]-[w(x+\mathrm{\Delta }x)-w(x)]\\ & =\mathrm{\Delta }u+\mathrm{\Delta }v-\mathrm{\Delta }w,\end{array}$ ndërsa raporti i kësaj shtese me shtesë e argumentit: $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }y}=\frac{\mathrm{\Delta }u+\mathrm{\Delta }v-\mathrm{\Delta }w}{\mathrm{\Delta }x}\frac{\mathrm{\Delta }u}{\mathrm{\Delta }x}+\frac{\mathrm{\Delta }v}{\mathrm{\Delta }x}-\frac{\mathrm{\Delta }w}{\mathrm{\Delta }x}.$         Vlera kufitare e këtij raporti, kur $\mathrm{\Delta }x\to 0$, shprehet:         $\begin{array}{cc}{f}^{\prime }(x)& =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}(\frac{\mathrm{\Delta }u}{\mathrm{\Delta }x}+\frac{\mathrm{\Delta }v}{\mathrm{\Delta }x}-\frac{\mathrm{\Delta }w}{\mathrm{\Delta }x})=\\ & =\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }u}{\mathrm{\Delta }x}+\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }v}{\mathrm{\Delta }x}-\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }w}{\mathrm{\Delta }x}=u^{\prime }(x)+v^{\prime }(x)-w^{\prime }(x)\end{array}$

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