Question

Develop a rule for \(D_{x}[f(x) g(x) h(x)]\)

Short answer

The rule: \( D_x[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \).

Step-by-step solution

Understand the Goal

We need to find the derivative of the product of three functions, specifically \( f(x), g(x), \) and \( h(x) \). This involves applying differentiation rules that extend the product rule for more than two functions.

Recall the Product Rule for Two Functions

The derivative of the product of two functions \( D_x[u(x)v(x)] \) is given by \( u'(x)v(x) + u(x)v'(x) \). We'll generalize this for three functions.

Apply the Product Rule to Three Functions

When differentiating the product of three functions \( D_x[f(x)g(x)h(x)] \), treat it as the product of \( f(x) \) and the product of \( g(x)h(x) \). First, use the product rule on \( f(x) \) and \( g(x)h(x) \): \[ D_x[f(x)g(x)h(x)] = f'(x) imes (g(x)h(x)) + f(x) imes D_x[g(x)h(x)] \]

Differentiate the Remaining Product

Apply the product rule again to \( g(x)h(x) \) using its formula:\[ D_x[g(x)h(x)] = g'(x)h(x) + g(x)h'(x) \].Substitute this back into the expression from Step 3.

Combine and Simplify

After substituting, the derivative becomes:\[ D_x[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)(g'(x)h(x) + g(x)h'(x)) \].Expand and simplify to:\[ f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \].

State the General Rule

The derivative of a product of three functions \( F(x) = f(x)g(x)h(x) \) is:\[ D_x[F(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \].Each term corresponds to differentiating one function while keeping the other two constant.

Key concepts

Product Rule for Multiple Functions

When dealing with the differentiation of multiple functions multiplied together, such as in the expression \( D_x[f(x)g(x)h(x)] \), the Product Rule becomes a vital tool. The classic Product Rule in calculus is widely used for differentiating the product of two functions. It states that if you have two functions \( u(x) \) and \( v(x) \), their derivative is given by:\[ D_x[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). \]This formula shows the sum of products, where for each term, one function is differentiated while the other remains unchanged.
The challenge arises when extending this principle to more than two functions. For three functions, \( f(x), g(x), \text{ and } h(x) \), the rule adapts by applying the Product Rule iteratively. First, treat two of the functions as a single entity and differentiate accordingly. For \( f(x)g(x)h(x) \), consider \( f(x) \) paired with the product \( g(x)h(x) \), then apply the Product Rule:\[ D_x[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)D_x[g(x)h(x)]. \]
This approach is scalable to any number of functions, making it a powerful technique in multivariable differentiation.

Calculus Fundamentals

Calculus is a branch of mathematics focused on limits, derivatives, integrals, and infinite series. It plays a crucial role in understanding changes described by mathematical functions. There are two main types of calculus: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which represents how a function changes at any point.
The derivative of a function measures the rate at which the function's value changes as its input changes. This is invaluable for analyzing and predicting behaviors in a myriad of scientific fields, including physics, engineering, and economics. Calculus allows us to solve complex real-world problems by modeling situations where things change in a dynamic way.
Thus, learning to apply calculus, especially differentiation, is foundational for grasping the nuances of mathematical analysis.

Differentiation Techniques

Differentiation is the process of finding a derivative, and it comes with a variety of techniques allowing us to tackle different types of functions and combinations. When you start learning differentiation, you encounter basic rules like the power rule, the product rule, and the chain rule. Each of these serves a specific purpose and simplifies finding derivatives of complex functions.
- **Power Rule**: Provides a quick way to differentiate polynomial functions. For any function \( f(x) = x^n \), its derivative is \( f'(x) = nx^{n-1} \).
- **Product Rule**: As explained in earlier sections, this is used for functions that are products of multiple terms. It ensures each piece is differentiated relative to its partners.
- **Chain Rule**: Helps differentiate composite functions, providing a method to handle functions within functions.
Understanding these techniques is crucial. They are the tools that help dissect and analyze mathematical expressions, providing insights into how changes in variables affect outcomes.