Final answer:
To differentiate y = x² sin(x), we apply the product rule to obtain the derivative, which is y' = 2x sin(x) + x² cos(x).
Step-by-step explanation:
To differentiate the function y = x² sin(x) using the product rule, we need to apply the rule which states that if we have two functions u(x) and v(x), the derivative of their product y = u(x)v(x) is given by y' = u'(x)v(x) + u(x)v'(x). In our case, u(x) = x² and v(x) = sin(x).
Taking the derivatives of both u and v gives us u'(x) = 2x and v'(x) = cos(x). Hence applying the product rule:
y' = u'(x)v(x) + u(x)v'(x) = (2x)(sin(x)) + (x²)(cos(x)).
Therefore, the derivative of the function y = x² sin(x) is y' = 2x sin(x) + x² cos(x).
To differentiate y = x² sin(x), we apply the product rule to obtain the derivative, which is y' = 2x sin(x) + x² cos(x).