Answer:
[tex] \displaystyle f'(x) = 3x^2 {e}^{ {x}^{3} } [/tex]
Step-by-step explanation:
we would like to figure out the first derivative of the following:
[tex] \displaystyle f(x) = {e}^{x ^{3} } [/tex]
to do so take derivative In both sides:
[tex] \displaystyle f'(x) = \frac{d}{dx} {e}^{x ^{3} } [/tex]
to differentiate the above we can consider composite function derivation given by
[tex] \rm\displaystyle \frac{d}{dx} f(g(x)) = \frac{d}{dx} f'(g(x)) \times \frac{d}{dx} g'(x)[/tex]
let
g(x)=u
so we obtain:
[tex] \displaystyle f'(x) = \frac{d}{dx} {e}^{u} \times \frac{d}{dx} u[/tex]
substitute back:
[tex] \displaystyle f'(x) = \frac{d}{dx} {e}^{ {x}^{3} } \times \frac{d}{dx} {x}^{3} [/tex]
by using derivation rule we acquire:
[tex] \displaystyle f'(x) = 3x^2 {e}^{ {x}^{3} } [/tex]
and we are done!
