Calculus I, II and III

The differentiation uv rule, commonly known as the product rule, is a fundamental technique in calculus for finding the derivative of the product of two functions. The rule states that if you have two differentiable functions \( u(x) \) and \( v(x) \), then the derivative of their product \( u(x)v(x) \) with respect to \( x \) is given by: \[ \frac{d}{dx}[u(x)v(x)] = u(x) \frac{dv(x)}{dx} + v(x) \frac{du(x)}{dx} \] Here's a step-by-step guide on how to apply the product rule: 1. **Identify the Functions**: First, identify the two functions that are being multiplied together, which we will call \( u(x) \) and \( v(x) \). 2. **Differentiate Each Function Separately**: Compute the derivative of \( u(x) \) with respect to \( x \), denoted as \( \frac{du(x)}{dx} \), and the derivative of \( v(x) \) with respect to \( x \), denoted as \( \frac{dv(x)}{dx} \). 3. **Apply the Product Rule**: Multiply \( u(x) \) by the derivative of \( v(x) \), and \( v(x) \) by the derivative of \( u(x) \), then add the two products together. 4. **Simplify**: If possible, simplify the resulting expression to obtain the final derivative of the product. Let's apply the product rule to an example: Suppose we want to differentiate the function \( f(x) = x^2 \sin(x) \). Here, \( u(x) = x^2 \) and \( v(x) = \sin(x) \). Step 1: Identify the functions. \[ u(x) = x^2 \] \[ v(x) = \sin(x) \] Step 2: Differentiate each function separately. \[ \frac{du(x)}{dx} = 2x \] \[ \frac{dv(x)}{dx} = \cos(x) \] Step 3: Apply the product rule. \[ \frac{d}{dx}[u(x)v(x)] = u(x) \frac{dv(x)}{dx} + v(x) \frac{du(x)}{dx} \] \[ \frac{d}{dx}[x^2 \sin(x)] = x^2 \frac{d}{dx}[\sin(x)] + \sin(x) \frac{d}{dx}[x^2] \] \[ \frac{d}{dx}[x^2 \sin(x)] = x^2 \cos(x) + \sin(x) \cdot 2x \] Step 4: Simplify the expression. \[ \frac{d}{dx}[x^2 \sin(x)] = x^2 \cos(x) + 2x \sin(x) \] Therefore, the derivative of \( f(x) = x^2 \sin(x) \) with respect to \( x \) is \( x^2 \cos(x) + 2x \sin(x) \).

Question

Apply the differentiation uv rule (product rule) and provide an example.

Solution

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