Final answer:
The derivative of the function f(t) = t·cos(t) + t^2·sin(t) is found using product rule, resulting in f'(t) = cos(t) + t·sin(t) + t^2·cos(t).
Step-by-step explanation:
To find the derivative of the function f(t) = t·cos(t) + t2·sin(t), we need to use the product rule and the chain rule for differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function.The first function multiplied by the derivative of the second function.
Applying the product rule: Derivative of t·cos(t): t'·cos(t) + t·(-sin(t)). Derivative of t2·sin(t): 2t·sin(t) + t2·cos(t). Next, combine the results to get the total derivative of the function: cos(t) - t·sin(t) + 2t·sin(t) + t2·cos(t). Simplifying, the derivative f'(t) is: cos(t) + (2t - t)·sin(t) + t2·cos(t) Which simplifies further to: f'(t) = cos(t) + t·sin(t) + t2·cos(t)