H-factor Method for differentation by using first principle
Example 1 (h-factor method)
Find $\displaystyle f'(x)$, if $f(x)=\sqrt{x}.$
\begin{eqnarray*}
f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\
&=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(x+h)-x}\\
&=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h})^2-(\sqrt{x})^2}\\
&=&\lim_{h\to 0}
\frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}\\
&=& \lim_{h\to 0}
\frac{1}{(\sqrt{x+h}+\sqrt{x})}\\
&=& \frac{1}{\sqrt{x}+\sqrt{x}}\\
&=&\frac{1}{2\sqrt{x}}
\end{eqnarray*}
Example 2 (h-factor method)
Find $\displaystyle f'(x)$, if $f(x)=\sqrt[3]{x}.$
\begin{eqnarray*}
f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\
&=&\lim_{h\to 0}\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{(x+h)-x}\\
&=&\lim_{h\to 0}\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{(\sqrt[3]{x+h})^3-(\sqrt[3]{x})^3}\\
&=&\lim_{h\to 0}
\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{(\sqrt[3]{x+h}-\sqrt[3]{x})((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \lim_{h\to 0}
\frac{1}{((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \frac{1}{\sqrt[3]{x^2}+\sqrt[3]{x^2} +\sqrt[3]{x^2}}\\
&=&\frac{1}{3\sqrt[3]{x^2}}
\end{eqnarray*}
Example 3 (h-factor method)
Find $\displaystyle f'(x)$, if $f(x)=\sqrt[3]{x^2}.$
\begin{eqnarray*}
f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\
&=&\lim_{h\to 0}\frac{(\sqrt[3]{x+h})^2-(\sqrt[3]{x})^2}{(x+h)-x}\\
&=&\lim_{h\to 0}\frac{(\sqrt[3]{x+h}-\sqrt[3]{x})(\sqrt[3]{x+h}+\sqrt[3]{x})}{(\sqrt[3]{x+h})^3-(\sqrt[3]{x})^3}\\
&=&\lim_{h\to 0}
\frac{(\sqrt[3]{x+h}-\sqrt[3]{x})(\sqrt[3]{x+h}+\sqrt[3]{x})}{(\sqrt[3]{x+h}-\sqrt[3]{x})((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \lim_{h\to 0}
\frac{\sqrt[3]{x+h}+\sqrt[3]{x}}{((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \frac{\sqrt[3]{x}+\sqrt[3]{x}}{\sqrt[3]{x^2}+\sqrt[3]{x^2} +\sqrt[3]{x^2}}\\
&=&\frac{2\sqrt[3]{x}}{3\sqrt[3]{x^2}}=\frac{2}{3\sqrt[3]{x}}
\end{eqnarray*}
Example 4 (h-factor method)
Find $\displaystyle f'(x)$, if $f(x)=\sqrt[3]{x^4}.$
\begin{eqnarray*}
f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\
&=&\lim_{h\to 0}\frac{(\sqrt[3]{x+h})^4-(\sqrt[3]{x})^4}{(x+h)-x}\\
&=&\lim_{h\to 0}\frac{((\sqrt[3]{x+h})^2-(\sqrt[3]{x})^2)((\sqrt[3]{x+h})^2+(\sqrt[3]{x})^2)}{(\sqrt[3]{x+h})^3-(\sqrt[3]{x})^3}\\
&=&\lim_{h\to 0}
\frac{(\sqrt[3]{x+h}-\sqrt[3]{x})(\sqrt[3]{x+h}+\sqrt[3]{x})((\sqrt[3]{x+h})^2+(\sqrt[3]{x})^2)}{(\sqrt[3]{x+h}-\sqrt[3]{x})((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \lim_{h\to 0}
\frac{(\sqrt[3]{x+h}+\sqrt[3]{x})((\sqrt[3]{x+h})^2+(\sqrt[3]{x})^2)}{((\sqrt[3]{x+h})^2+
\sqrt[3]{x+h}\sqrt[3]{x}
+(\sqrt[3]{x})^2)}\\
&=& \frac{(\sqrt[3]{x}+\sqrt[3]{x})((\sqrt[3]{x})^2+(\sqrt[3]{x})^2)}{\sqrt[3]{x^2}+\sqrt[3]{x^2} +\sqrt[3]{x^2}}\\
&=&\frac{2\sqrt[3]{x}\times2\sqrt[3]{x^2}}{3\sqrt[3]{x^2}}=\frac{4\sqrt[3]{x}}{3}
\end{eqnarray*}
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