Generalizing the Product Rule The Derivative Product Rule gives the formula


d/dx (uv) = u (dv/dx) + (du/dx) v


for the derivative of the product uv of two differentiable functions of x.


b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?

Verified step by step guidance

1. To find the derivative of the product of four functions, u₁u₂u₃u₄, we can extend the product rule. The product rule for two functions states that the derivative of a product uv is given by d/dx (uv) = u (dv/dx) + (du/dx) v.
2. For four functions, we apply the product rule iteratively. First, consider the product of the first two functions, u₁u₂, and the product of the last two functions, u₃u₄. We can treat these as two separate products and apply the product rule to each.
3. The derivative of u₁u₂u₃u₄ can be expressed as: d/dx (u₁u₂u₃u₄) = (d/dx (u₁u₂)) u₃u₄ + u₁u₂ (d/dx (u₃u₄)).
4. Now, apply the product rule to each part: d/dx (u₁u₂) = u₁ (du₂/dx) + (du₁/dx) u₂ and d/dx (u₃u₄) = u₃ (du₄/dx) + (du₃/dx) u₄.
5. Substitute these derivatives back into the expression: d/dx (u₁u₂u₃u₄) = [u₁ (du₂/dx) + (du₁/dx) u₂] u₃u₄ + u₁u₂ [u₃ (du₄/dx) + (du₃/dx) u₄].

Key Concepts

Product Rule

The Product Rule is a fundamental theorem in calculus that provides a method for differentiating the product of two functions. It states that the derivative of the product of two functions u and v is given by d/dx(uv) = u(dv/dx) + v(du/dx). This rule can be extended to more than two functions, which is essential for solving problems involving multiple products.

Higher-Order Product Rule

The Higher-Order Product Rule generalizes the Product Rule to the differentiation of products involving more than two functions. For three functions, the derivative can be expressed as d/dx(u₁u₂u₃) = u₁(u₂' u₃ + u₂ u₃') + u₂(u₁' u₃ + u₁ u₃') + u₃(u₁' u₂ + u₁ u₂'). This pattern continues, allowing for systematic differentiation of products of any number of functions.

Differentiability

Differentiability is a key concept in calculus that indicates whether a function has a derivative at a given point. A function is differentiable at a point if it is continuous there and its derivative exists. This property is crucial when applying the Product Rule, as it ensures that the functions involved can be differentiated, allowing for the application of the rule to find the derivative of their product.