Question

Find the derivative of the function. $$ f(t)=t^{2 / 3}-t^{1 / 3}+4 $$

Short answer

The derivative of the function \(f(t)\) is \(f'(t) = 2/3 * t^{-1/3} - 1/3 * t^{-2/3}.\)

Step-by-step solution

Apply Power Rule to the first term

The first term of the function is \(t^{2/3}\). The power rule states that the derivative of \(t^n\) is \(n*t^{n-1}\) where n is the power. Apply the power rule to the first term: \(2/3 * t^{2/3 - 1} = 2/3 * t^{-1/3}.\)

Apply Power Rule to the second term

The second term of the function is \(-t^{1/3}\). Again, apply the power rule: \(-1/3 * t^{1/3 - 1} = -1/3 * t^{-2/3}.\)

Derivative of constant term

The third term of the function is a constant, 4. The derivative of a constant is always 0. Thus the derivative of the third term is 0.

Combine the results

By adding all calculated results, the derivative of function \(f(t)\) becomes \(f'(t) = 2/3 * t^{-1/3} - 1/3 * t^{-2/3}.\)