Differentiate each of the following from first principle: `x^2e^x`

Explanation

` Let f(x)=x^{2} e^{x} `

`f(x+h)=(x+h)^{2} e^{(x+h)} `

`=lim _{h - 0} frac{f(x+h)-f(x)}{h} `

`=lim _{h - 0} frac{(x+h)^{2} e^{(x+h)}-x^{2} e^{x}}{h}`

`=lim _{h - 0}(frac{x^{2} e^{x+h}-x^{2} e^{x}}{h}+frac{2 x h e^{(x+h)}}{h}+frac{h^{2} e^{(x+h)}}{h}) `

`=lim _{h - 0}(frac{x^{2} e^{x}(e^{(x+h)-x}-1)}{h}+2 x e^{(x+h)}+h e^{(x+h)}) `

`=lim _{h - 0}(x^{2} e^{x} frac{(e^{h}-1)}{h}+2 x e^{(x+h)}+h e(x+h))`

`=x^{2} e^{x}+2 x e^{x}+0 x e^{x}[ lim _{x - 0} frac{e^{x}-1}{x}=1]`
`frac {d x}(x^{2} e^{x})=e^{x}(x^{2}+2 x)`