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

Question

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

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Answer

We have,$\text{f(x)}=\text{x}^2+\text{x}+3$
$\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\big(\text{x+h}\big)-\text{f}\big(\text{x}\big)}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\bigg\{{\big(\text{x+h}\big)^2+\big(\text{x+h}\big)+3}\bigg\}-\text{x}^2+\text{x}+3}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{x}^2+\text{h}^2+2\text{xh}+\text{x+h+3}-\text{x}^2-\text{x}-3}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{2\text{xh}+\text{h}^2+\text{h}}{\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}\big(2\text{x+h+1}\big)}{\text{h}}$
$=2\text{x}+0+1$
$=2\text{x}+1$

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