Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. $$ f(t)=-3 t^{2}+2 t-4 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(t) = -3t^2 + 2t - 4\) is \(f'(t) = -6t + 2\).

Step 1: Identify the Terms and Power Rule

The function given is a second-degree polynomial function, \(f(t) = -3t^2 + 2t - 4\). According to the Power Rule; the derivative of a function of the form \(t^n\) is given by \(n*t^{n-1}\). In this case, the power rule will be applied separately to each term in the function.

Step 2: Apply the Power Rule to Each Term

Apply the power rule to each term. For the first term, \( -3t^2 \), the power rule gives us \( -3*2*t^{2-1}\), which simplifies to \( -6t \). For the second term, \( 2t \), the power rule gives us \( 2*1*t^{1-1} = 2 \). The last term \( -4 \) is a constant and its derivative is zero.

Step 3: Combine the Derivatives

The derivatives found from each term in the function are combined to give the derivative of the entire function. This results in \(f'(t) = -6t + 2 + 0\), which simplifies to \(f'(t) = -6t + 2\).