Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The derivative is \( f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \).

Step 1: Identify the Product Rule

The product rule for differentiation states that if you have two functions, say \(u\) and \(v\), that are being multiplied, their derivative is given by \(D_x[u \, v] = u' \, v + u \, v'\). In essence, the derivative of a product is the derivative of the first function times the second plus the first function times the derivative of the second.

Step 2: Extend the Product Rule for Three Functions

Applying the product rule to three functions \(f(x), g(x),\) and \(h(x)\), we can extend the concept as follows. We need to account for each function's derivative in the product. We can think of the original product rule and extend:

Step 3: Apply the Product Rule Strategically

For three functions, \( D_x[f(x) \, g(x) \, h(x)] \), apply the product rule at each step. First, differentiate \(f(x)\), while considering \(g(x) \, h(x)\) as one entity. Likewise for \(g(x)\) and \(h(x)\): \[ D_x[f(x) \, g(x) \, h(x)] = f'(x) \, g(x) \, h(x) + f(x) \, g'(x) \, h(x) + f(x) \, g(x) \, h'(x) \]

Step 4: Summarize the Rule

The final derivative of the product of three functions \(f(x), g(x), \) and \(h(x)\) is given as: \[ D_x[f(x) \, g(x) \, h(x)] = f'(x) \, g(x) \, h(x) + f(x) \, g'(x) \, h(x) + f(x) \, g(x) \, h'(x) \].This works by differentiating each function one at a time while keeping the remaining two functions untouched.

Key Concepts

Differentiation

The process of differentiation is a fundamental concept in calculus. It involves finding the derivative of a function to determine how it changes with respect to a variable, typically denoted as \(x\). This process helps us understand the rate of change of a function.Differentiation is commonly used:

• To calculate the slope of a tangent line to a curve at any point.
• To find how one quantity changes concerning another.

Applying the rules of differentiation, like the product rule, allows us to find derivatives of more complex functions. These rules help simplify and systematically approach these tasks.Differentiating products of functions, as in our exercise, requires special attention to the product rule, which makes it easier to handle these operations without direct involvement in multiplying and then differentiating each time.

Derivative

The derivative is a measure of how a function changes as its input changes. In mathematical terms, it's the limit of the rate of change of the function values as the change in the input approaches zero. In essence:\[ \text{derivative of a function } f(x) = f'(x) \]The derivative shows:

• The rate at which a function increases or decreases.
• The behavior of the function at certain points, such as the slope at a particular point of its graph.

In the context of our exercise, the derivative is applied to products of multiple functions. It's crucial to apply the product rule properly to find the derivatives of these products. For three functions, you derive each one individually, while maintaining the others unchanged, to form an expression that respects the change of each derivative.

Three Functions

Handling the differentiation of three functions, such as \(f(x), g(x), \) and \(h(x)\), involves an extension of the basic product rule. The task is to derive a rule that accommodates the change in all three functions when they are multiplied together.Using the product rule, differentiation for three functions aims to find:\[D_x[f(x) \cdot g(x) \cdot h(x)] = f'(x) \cdot g(x) \cdot h(x) + f(x) \cdot g'(x) \cdot h(x) + f(x) \cdot g(x) \cdot h'(x)\]This formula entails:

• Differentiating each function one at a time.
• Keeping the other two functions unaltered while differentiating.
• Summing up all these differentiated terms.

By applying this extended product rule, you gain a comprehensive understanding of how the differentiation process accommodates multiple changing rates within a single product.