Question

Use the General Power Rule to find the derivative of the function. $$ f(t)=(9 t+2)^{2 / 3} $$

Short answer

\(f'(t) = 6(9t + 2)^{-1/3}\)

Step-by-step solution

Find the derivative of u

First, differentiate the function inside the brackets, \(u = 9t + 2\). The derivative of a constant times a variable is simply the constant, so the derivative of 9t is 9. The derivative of a constant is 0, so the derivative of 2 is 0. So, \(u' = 9 + 0 = 9\).

Apply the General Power Rule

Now, we can apply the General Power Rule for differentiation, which states that the derivative of \(u^n\) is \(n.u^(n-1).u'\). Plugging in the values in the formula, this becomes \( (2/3).(9t + 2)^(2/3 - 1).9 \) = \( (2/3).(9t + 2)^(-1/3).9 \).

Simplify the expression

The last step is to simplify the expression. The final expression of derivative becomes \(2/3 . 9(9t + 2)^(-1/3) = 6(9t + 2)^(-1/3)\).