Derivatives — MATHS STD 11 Science — Question

$\text{f(x)}=\text{x}^2+\text{x}+3$

$\text{f}'\text{(x)}=\lim\limits_{\text{h}\rightarrow0}\frac{\text{f}\big(\text{x+h}\big)-\text{f}\big(\text{x}\big)}{\text{h}}$

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

$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{x}^2+\text{h}^2+2\text{xh}+\text{x+h+3}-\text{x}^2-\text{x}-3}{\text{h}}$

$=\lim\limits_{\text{h}\rightarrow0}\frac{2\text{xh}+\text{h}^2+\text{h}}{\text{h}}$

$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}\big(2\text{x+h+1}\big)}{\text{h}}$

$=2\text{x}+0+1$

$=2\text{x}+1$