H-factor Method (Differentiation)

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*}