Differentiate the following from first principle:(x+2)^3 - Vidyadip

Question

Differentiate the following from first principle:$(\text{x}+2)^3$

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Answer

We have,$\text{f(x)}=\text{(x+2)}^3$
$\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\big(\text{x+h}\big)-\text{f}\big(\text{x}\big)}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\big(\text{x}+\text{h}+2\big)^3-\big(\text{x+2}\big)^3}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\bigg\{\big(\text{x+2}\big)+\text{h}\bigg\}^3-\big(\text{x+h}\big)^3}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\big(\text{x+2}\big)^3+\text{h}^3+3\text{h}\big(\text{x+2}\big)^2+3\big(\text{x+2}\big)\text{h}^2-\big(\text{x+2}\big)^3}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}3\big(\text{x+2}\big)^2+3\big(\text{x+2}\big)\text{h+h}^2$
$=3\big(\text{x+2}\big)^2$

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