Question

Calcule df/dt, sendo f(t)= cos(e^t^3-1)

Answer 1

$\boxed{f(t)=\cos{\big(e^{t^3-1}\big)}}}$

Para calcular a derivada de f(t) em função de t, devemos utilizar a Regra da Cadeia:

$\boxed{f(g(h(\dots(x))))=f'(g(h(\dots(x))))[g'(h(\dots(x)))][h'(\dots(x))]\dots}$

Dessa forma:

$\dfrac{df}{dt}=\dfrac{d}{dt}\bigg(\cos{\big(e^{t^3-1}\big)}\bigg)\bigg[\dfrac{d}{dt}\big(e^{t^3-1}\big)\bigg]\bigg[\dfrac{d}{dt}\big(t^3-1\big)\bigg]\ \therefore$

$\dfrac{df}{dt}=\bigg(-\sin{\big(e^{t^3-1}\big)}\bigg)\bigg[e^{t^3-1}\bigg]\bigg[3t^2\bigg]\ \therefore$

$\boxed{\dfrac{df}{dt}=-3t^2e^{t^3-1}\sin{\big(e^{t^3-1}\big)}}$