Question

Find the derivative in each case. You need not simplify your answer.

a. f(t)=(-3t²+1/∛(4t))(t³+2∜t)
b. g(t)=(∛t+5)/(2∜t)

Answer 1

To solve both equations, we use the product rule for the first and the quotient rule for the second without simplifying the results to obtain the derivatives.

Step-by-step explanation:

To find the derivative in each case without simplifying the answer, we will use the product rule for derivatives and the quotient rule where applicable.

Part a: f(t) = (-3t² + 1/∛(4t))(t³ + 2∜¹t)

To differentiate f(t), we apply the product rule, which states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function:

f'(t) = d/dt[-3t² + 1/∛(4t)] * (t³ + 2∜¹t) + [-3t² + 1/∛(4t)] * d/dt[t³ + 2∜¹t]

Part b: g(t) = (∛t + 5)/(2∜¹t)

For g(t), we utilize the quotient rule which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator:

g'(t) = d/dt[∛t + 5] * 2∜¹t - (∛t + 5) * d/dt[2∜¹t] / (2∜¹t)²