Question

Find the first derivative of the following:
f(x)=e^{x³}​

Answer 1

`\displaystyle f'(x) = 3x^2 {e}^{ {x}^(3) }`

Explanation:

we would like to figure out the first derivative of the following:

`\displaystyle f(x) = {e}^{x ^(3) }`

to do so take derivative In both sides:

`\displaystyle f'(x) = (d)/(dx) {e}^{x ^(3) }`

to differentiate the above we can consider composite function derivation given by

`\rm\displaystyle (d)/(dx) f(g(x)) = (d)/(dx) f'(g(x)) * (d)/(dx) g'(x)`

let

g(x)=u

so we obtain:

`\displaystyle f'(x) = (d)/(dx) {e}^(u) * (d)/(dx) u`

substitute back:

`\displaystyle f'(x) = (d)/(dx) {e}^{ {x}^(3) } * (d)/(dx) {x}^(3)`

by using derivation rule we acquire:

`\displaystyle f'(x) = 3x^2 {e}^{ {x}^(3) }`

and we are done!

Answer 2

`answer = 3 {x}^(2) {e}^{ {x}^(3) }`