Derivatives Find and simplify the derivative of the following functions.
s(t) = t⁴/³ / e^t

Verified step by step guidance

1. Step 1: Identify the function s(t) = \frac{t^{4/3}}{e^t}. This is a quotient of two functions, so we will use the Quotient Rule to find the derivative.
2. Step 2: Recall the Quotient Rule: If you have a function \frac{u(t)}{v(t)}, its derivative is \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}. Here, u(t) = t^{4/3} and v(t) = e^t.
3. Step 3: Differentiate u(t) = t^{4/3}. Use the power rule: \frac{d}{dt}[t^n] = nt^{n-1}. So, u'(t) = \frac{4}{3}t^{1/3}.
4. Step 4: Differentiate v(t) = e^t. The derivative of e^t with respect to t is simply e^t, so v'(t) = e^t.
5. Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the Quotient Rule formula: \frac{\frac{4}{3}t^{1/3}e^t - t^{4/3}e^t}{(e^t)^2}. Simplify the expression by factoring and combining like terms.