find the derivative of the following function.\nf(t) = t^{\frac{8}{5}}e^{t}\nf(t) = square

Explanation:

Step1: Apply product - rule

The product - rule states that if (y = u\cdot v), then (y'=u'v + uv'). Let (u = t^{\frac{8}{5}}) and (v = e^{t}).

Step2: Find the derivative of (u)

Using the power - rule ((x^{n})'=nx^{n - 1}), for (u=t^{\frac{8}{5}}), we have (u'=\frac{8}{5}t^{\frac{8}{5}-1}=\frac{8}{5}t^{\frac{3}{5}}).

Step3: Find the derivative of (v)

The derivative of (v = e^{t}) is (v'=e^{t}).

Step4: Calculate (f'(t))

By the product - rule (f'(t)=u'v+uv'), substituting (u), (u'), (v), and (v') we get (f'(t)=\frac{8}{5}t^{\frac{3}{5}}e^{t}+t^{\frac{8}{5}}e^{t}=t^{\frac{3}{5}}e^{t}(\frac{8}{5}+t)).

Answer:

(t^{\frac{3}{5}}e^{t}(\frac{8}{5}+t))