Problem 1 - Calculus Early Transcendentals

Question

How do you find the derivative of the product of two functions that are differentiable at a point?

Step-by-Step Solution

Answer

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.

Step 1: Identify the functions u(x) and v(x)

The given exercise asks to find the derivative of the product of two differentiable functions at a point. Let the functions be u(x) and v(x).

Step 2: Differentiate each function

Calculate the derivative of each function, u'(x) and v'(x). If u(x) and v(x) are differentiable functions, their derivatives can be easily computed using basic differentiation rules.

Step 3: Apply the Product Rule

Now that we have the derivatives u'(x) and v'(x), we can apply the Product Rule to find the derivative of their product. According to the Product Rule: (uv)' = u'v + uv' Substitute u'(x) and v'(x) into the formula: (uv)' = u'(x)v(x) + u(x)v'(x)

Step 4: Compute the derivative at the given point

After finding the general derivative of the product of u(x) and v(x), substitute the specified point into the expression to find the value of the derivative at that particular point.