Find the derivative of the following function from first principl… - Vidyadip

Question

Find the derivative of the following function from first principle. $\text{x}^3-27$

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Answer

Let $\text{f}(\text{x})=\text{x}^3-27$. Accordingly,from the first principle, $\text{f}'(\text{x})=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h})-\text{f}(\text{x})}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\big[(\text{x}+\text{h})^3-27\big]-(\text{x}^3-27)}{\text{h}}$ $\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}^3+3\text{x}^2\text{h}+3\text{x}\text{h}^2}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}{(\text{h}^2+3\text{x}^2+3\text{x}\text{h})}$$=0+3\text{x}^2+0=3\text{x}^2$

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