Differentiate `x e^x` from first principles.

Explanation

Let `f(x)=x e^{x}`, then `f(x+h)=(x+h) e^{x+h}`


`therefore f^{prime}(x)=lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h} `

`=lim _{h rightarrow 0} frac{(x+h) e^{x+h}-x e^{x}}{h}=lim _{h rightarrow 0} frac{x e^{x+h}-x e^{x}+h e^{x+h}}{h} `

`=lim _{h rightarrow 0} frac{x e^{x}(e^{h}-1)}{h}+lim _{h rightarrow 0} frac{h e^{x+h}}{h} `

`=x e^{x}+e^{x} [because lim _{h rightarrow 0} frac{e^{h-1}}{h}=1] `

`=e^{x}(x+1)`