Question

differentiate e^x^1+2 from first principle

Answer 1

Dear student

as per first principle, differentiation of function is defined as

$$\begin{gathered} f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ here \\ asstatedinquestione^x^1+2 \\ f\left(x\right)=e^x^1+2=e^{x^1}+2=e^x+2 \\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{\left(e^{x+h}+2\right)-(e^x+2)}{h} \\ =\underset{h\to 0}{\lim }\frac{\left(e^x{e}^h+2\right)-(e^x+2)}{h} \\ =\underset{h\to 0}{\lim }\frac{e^x{e}^h+2-e^x-2}{h} \\ =e^x\underset{h\to 0}{\lim }\frac{e^h-1}{h} \\ =e^x\underset{h\to 0}{\lim }\frac{e^h-0}{1}applyingLHospitalruleas0/0form \\ =e^x\times 1 \\ =e^x \end{gathered}$$

Regards