use the quotient rule to differentiate the following k(t)=e^3t/4-t^2 (-2<t<2)

To use the quotient rule to differentiate the function k(t) = e^(3t) / (4 - t^2), you can follow these steps:

1. Identify the numerator and denominator of the function. In this case, the numerator is e^(3t) and the denominator is (4 - t^2).

2. Apply the quotient rule, which states that for a function u(t) / v(t), the derivative is given by:
k'(t) = [u'(t)v(t) - u(t)v'(t)] / [v(t)]^2

3. Determine the derivatives of the numerator and denominator. Using the chain rule, we have:
- The derivative of e^(3t) with respect to t is (e^(3t))' = 3e^(3t)
- The derivative of (4 - t^2) with respect to t is (4 - t^2)' = -2t

4. Substitute these derivatives and the original functions into the quotient rule formula:
k'(t) = [(3e^(3t)(4 - t^2)) - (e^(3t)(-2t))] / [(4 - t^2)]^2

5. Simplify the expression:
k'(t) = (12e^(3t) - 3te^(3t) + 2te^(3t)) / (4 - t^2)^2
= (12e^(3t) - te^(3t)) / (4 - t^2)^2

Therefore, the derivative of k(t) = e^(3t) / (4 - t^2) is k'(t) = (12e^(3t) - te^(3t)) / (4 - t^2)^2.