91Ó°ÊÓ

Question: Given two differentiable functions u(x) = x^2 and v(x) = sin(x), find the derivative of their product at x = π. Answer: We will follow the steps from the solution above to find the derivative of the product of u(x) = x^2 and v(x) = sin(x) at x = π. Step 1: Identify the functions u(x) and v(x) u(x) = x^2 and v(x) = sin(x) Step 2: Differentiate each function u'(x) = 2x v'(x) = cos(x) Step 3: Apply the Product Rule (uv)' = u'v + uv' (uv)' = u'(x)v(x) + u(x)v'(x) (uv)' = (2x)(sin(x)) + (x^2)(cos(x)) Step 4: Compute the derivative at the given point (x = π) (uv)' = (2π)(sin(π)) + (π^2)(cos(π)) (uv)' = (2π)(0) + (π^2)(-1) (uv)' = -π^2 Thus, the derivative of the product of u(x) = x^2 and v(x) = sin(x) at x = π is -π^2.