Use the product rule to find the derivative of the following function.𝑓(𝑥)=𝑥2sin(𝑥)

Sure, let's use the product rule to find the derivative of the function f(x) = x^2*sin(x). The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Step 1: Identify the two functions that are being multiplied together. In this case, the two functions are u(x) = x^2 and v(x) = sin(x).

Step 2: Find the derivative of these two functions. The derivative of u(x) = x^2 is u'(x) = 2x. The derivative of v(x) = sin(x) is v'(x) = cos(x).

Step 3: Apply the product rule. According to the product rule, the derivative of f(x) = u(x)*v(x) is f'(x) = u(x)*v'(x) + v(x)u'(x). Substituting the functions and their derivatives gives us f'(x) = x^2cos(x) + sin(x)*2x.

So, the derivative of the function f(x) = x^2sin(x) is f'(x) = x^2cos(x) + 2x*sin(x).