Question

How do you find the 1st and 2nd derivative of `e^(x^2)`?

Answer 1

`f'(x)=2xe^(x^2),f''(x)=2e^(x^2)(2x^2+1)`

Explanation:

`color(orange)"Reminder"`

`d/dx(e^x)=e^x" and " d/dx(e^(g(x)))=e^(g(x)).g'(x)`

`color(blue)"First derivative"`

`f(x)=e^(x^2)rArrf'(x)=e^(x^2).2x=2xe^(x^2)`

`color(blue)"Second derivative"`

Differentiate using the `color(red)"product rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(f(x)=g(x)h(x)rArrf'(x)=g(x)h'(x)+h(x)g'(x))color(white)(a/a)|)))`

Differentiating `f'(x)=2xe^(x^2)`

now `g(x)=2xrArrg'(x)=2`

and `h(x)=e^(x^2)rArrh'(x)=2xe^(x^2)`

`rArrf''(x)=2x.2xe^(x^2)+2e^(x^2)=4x^2e^(x^2)+2e^(x^2)`

`=2e^(x^2)(2x^2+1)`