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.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Quotient Rule

The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function in the form f(t) = g(t) / h(t), the derivative is given by f'(t) = (g'(t)h(t) - g(t)h'(t)) / (h(t))². This rule is essential for differentiating functions like s(t) = t^(4/3) / e^t.

Exponential Functions

Exponential functions are functions of the form f(t) = a * e^(kt), where e is the base of the natural logarithm. These functions are characterized by their constant rate of growth or decay, making them crucial in various applications. Understanding how to differentiate exponential functions is vital when applying the Quotient Rule, especially when one of the functions in the ratio is an exponential.