Problem 75 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

True

Step 1: Understand the problem

We need to differentiate the function \(f(5x)\) with respect to \(x\). Then, we'll compare it with the expression \(5 \cdot f'(5x)\) to determine if they are equal.

Step 2: Differentiate the function

To differentiate \(f(5x)\) with respect to \(x\), apply the chain rule. The chain rule states: if \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). Here, \(u = 5x\), so \(\frac{du}{dx} = 5\) and \(y = f(u)\), hence \(\frac{dy}{du} = f'(u)\). Therefore, \(\frac{d}{dx} f(5x) = f'(5x) \cdot 5\).

Step 3: Compare with the given expression

The derivative we calculated is \(5 \cdot f'(5x)\), which is exactly the expression given in the problem. Therefore, the statement is true.

Key Concepts

Differentiation

Differentiation is a core concept in calculus used to determine the rate at which a function changes at any given point. In simple terms, it finds how the output of a function changes in response to changes in its input. Mathematically, it's about finding the derivative of a function.
When you differentiate a function like \( f(x) \), you find its derivative \( f'(x) \), which represents the slope or steepness of the function at any point \( x \). Differentiation is essential because it helps to understand behavior such as increasing and decreasing trends, and it’s crucial for optimizing functions in various fields such as physics, engineering, and economics.
For instance, if you have a function describing the distance traveled by a car over time, differentiating this function tells you the car's speed at any moment. Remember that differentiation follows specific rules, including the power rule, chain rule, product rule, and quotient rule, which simplify the process of finding derivatives.

Calculus Problem Solving

Solving calculus problems often involves several systematic steps to understand and solve them efficiently. Here’s a breakdown of a simple approach:

• **Understand the Problem**: Read the problem carefully to determine what is given and what needs to be found. Identify the type of calculus problem - differentiation, integration, or others.
• **Apply Relevant Formulas or Theorems**: Use specific rules and formulas suited to solve the problem, such as the chain rule for differentiating complex functions or integration by parts if you’re working with integrals.
• **Execute the Calculations**: Carry out the calculations step-by-step, ensuring each part of the process is clear and follows logically from the last.
• **Verify Your Solution**: Check your solution by substituting it back into the original function or problem to see if it makes sense or if it satisfies the conditions given.

Practicing a structured approach to problems helps to manage more advanced exercises easily and ensures you don’t miss important steps. Problem-solving in calculus requires both analytical thinking and practicing a lot.

Function Derivatives

Function derivatives are a way of expressing the function’s rate of change. When we talk about a derivative of a function \( f \), denoted \( f'(x) \) or \( \frac{df}{dx} \), we are describing how the function value changes as \( x \) changes.
Understanding this helps us to predict and analyze the behavior of functions more thoroughly.
When dealing with composite functions (like \( f(5x) \)), the chain rule is crucial in finding derivatives. The chain rule is an essential principle which allows for the differentiation of composed functions by breaking them down into simpler parts. In this case, it calculates derivatives in steps: first by differentiating the outer function with respect to its inner function, and then multiplying by the derivative of the inner function itself. In our exercise, using the chain rule correctly is what confirmed that the expression \( \frac{d}{dx} f(5x) = 5 \cdot f'(5x) \) is true. Function derivatives give insights into various characteristics of the function, including points of maxima and minima, inflection points, and overall trends. They are used extensively across disciplines for modeling and analysis tasks.