Final answer:
To find the derivative of the function f(t) = t³ cos(t) sin(t), assuming this is the corrected function, we would use the product rule and the chain rule, applying them iteratively to account for the multiple functions being multiplied together. The final derivative will include terms with t², cos(t), sin(t), and their respective derivatives.
Step-by-step explanation:
The function in question, f(t) = t cos(t) t² sin(t), seems to contain a typo or may not be written correctly. Assuming the intended function is f(t) = t³ cos(t) sin(t), which is a product of three functions, we can find the derivative using the product rule and the chain rule. The product rule states that the derivative of a product of two functions is given by first times the derivative of the second plus the second times the derivative of the first. In this case, however, we have a product of three functions, so we’ll need to apply the product rule multiple times.
Apply the product rule first to t³ and cos(t), and treat sin(t) as a third function to be multiplied later. The derivative of t³ is 3t², and the derivative of cos(t) is -sin(t). Applying the product rule to these two gives us the intermediate function: 3t² cos(t) - t³ sin(t). We then consider this as one function and multiply it by sin(t). Applying the product rule again, we find the derivative of f(t). Remember to apply the chain rule as needed for the terms involving sin(t) and cos(t).
The final derivative would thus involve terms with t², cos(t), and sin(t), as well as their respective derivatives.