Problem 78 - Brief Applied Calculus

Question

True or False: \(\frac{d}{d x} f(x+5)=f^{\prime}(x+5)\)

Step-by-Step Solution

Answer

True, the derivative is \(f'(x+5)\).

Step 1: Understanding the Derivative

To solve the exercise, we need to understand what the expression \(\frac{d}{dx} f(x+5)\) represents. This expression is asking us to find the derivative of the function \(f(x+5)\) with respect to \(x\).

Step 2: Using the Chain Rule

When taking the derivative of a function like \(f(x+5)\), the chain rule is applicable. The chain rule states that if we have a composite function \(f(g(x))\), then \(\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\). Here, \(g(x) = x+5\).

Step 3: Deriving g(x)

Since \(g(x) = x+5\), we find \(g'(x)\). The derivative of \(x + 5\) with respect to \(x\) is \(1\), as the derivative of \(x\) is \(1\) and the derivative of a constant is \(0\). So, \(g'(x) = 1\).

Step 4: Applying the Chain Rule

Using the chain rule, the derivative \(\frac{d}{dx} f(x+5)\) becomes \(f'(x+5) \cdot g'(x)\). Substituting \(g'(x) = 1\), we have \(f'(x+5) \cdot 1 = f'(x+5)\).

Step 5: Conclusion

Since \(\frac{d}{dx} f(x+5) = f'(x+5)\) after applying the chain rule, the statement is true.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus. It measures the rate at which a function changes as its input changes. Think of it as a tool that tells us how steep a curve is at any given point. In simpler terms, it's like observing the speed of a car at any moment during a trip. Derivatives help us find the slope of a tangent line to the graph of a function.

• The notation \(\frac{d}{dx} f(x)\) is used to denote the derivative of the function \(f\) with respect to \(x\).
• If \(f(x)\) represents some curve on a graph, its derivative \(f'(x)\) gives us the slope at any point \(x\).
• This slope tells us how fast and in what direction the function \(f\) is changing.

Derivatives make it possible to understand and predict trends in both mathematical functions and real-world phenomena. They are used in physics for calculating velocities and accelerations and in economics for determining profit maxima or minima.

Composite Function

A composite function is created when one function is applied to the result of another function. For example, if we have two functions \(f(x)\) and \(g(x)\), the composite function \(f(g(x))\) applies \(g\) first and then \(f\) to the result. This type of function often requires specific techniques for differentiation.

• In the given exercise, \(f(x+5)\) is a composite function where \(g(x) = x + 5\) and \(f\) is applied afterward.
• A composite function offers a structured yet complex relationship between input \(x\) and output \(f(g(x))\).

Understanding the concept of composite functions is crucial because they often appear in advanced mathematical problems. It requires recognizing that we are not dealing with a simple equation, but rather a combination of two or more functions.

Differentiation

Differentiation is the process of finding a derivative. It's the mathematical way of determining how a function changes as its input changes. This process involves applying rules and techniques to break down complex functions into manageable parts for analysis.

• The chain rule is a specific technique in differentiation used for finding derivatives of composite functions.
• In the exercise, we use the chain rule to differentiate \(f(x+5)\), recognizing that \(g(x) = x + 5\) and finding its derivative, \(g'(x)\).
• Differentiation transforms complicated mathematical tasks into simpler ones by using techniques like the product rule, quotient rule, and chain rule, each applying to different situations.

The essence of differentiation in calculus is to enable accurate predictions of changes and behaviors of functions in various fields, whether in natural sciences or engineering.