Final answer:
To differentiate the function g(t) = cos(t)sin(t), the product rule is used, resulting in the derivative g'(t) = -sin^2(t) + cos^2(t).
Step-by-step explanation:
The student is asking how to differentiate the function g(t) = cos(t)sin(t) using the product rule. The product rule is a derivative rule that allows us to differentiate functions that are products of two other functions. According to the product rule, if we have two functions u(t) and v(t), the derivative of their product u(t)v(t) is given by u'(t)v(t) + u(t)v'(t). Applying the product rule to our specific function, we let u(t) = cos(t) and v(t) = sin(t). This gives us:
- u'(t) = -sin(t), since the derivative of cos(t) is -sin(t)
- v'(t) = cos(t), since the derivative of sin(t) is cos(t)
Thus, the derivative g'(t) of the function g(t) is:
g'(t) = u'(t)v(t) + u(t)v'(t)
Which means:
g'(t) = (-sin(t))(sin(t)) + (cos(t))(cos(t))
After applying the product rule, we simplify to get:
g'(t) = -sin2(t) + cos2(t)