Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Find the value of the derivative of the function at the given point. $$ f(t)=4-\frac{4}{3 t} \quad\left(\frac{1}{2}, \frac{4}{3}\right) $$

Step-by-Step Solution

Answer

The value of the derivative of the function at the given point is 16.

Step 1 - Identify the function

Our function is \(f(t)=4-\frac{4}{3t}\)

Step 2 - Apply power rule for differentiation

We can rewrite the function as \(f(t) = 4 - 4 * (3t)^{-1}\). Now differentiate the function with respect to t, using power rule for differentiation, which is \((u^n)'=n*u^{n-1}*u'\). So, \(f'(t)=0 - (-1)*4*(3t)^{-2}*3\). Simplify the calculation, we get \(f'(t)=\frac{4}{t^2}\).

Step 3 - Substitute the value

Now replace t with the value at which the derivative is required. In the function \(f'(t)=\frac{4}{t^2}\), plug in t=\frac{1}{2} to get \(f'(\frac{1}{2})=\frac{4}{(\frac{1}{2})^2}=16\).