Problem 81
Question
Let \(f\) and \(g\) be differentiable functions of \(x\). Assume that denominators are not zero. True or False: \(\frac{d}{d x}(x \cdot f)=f+x \cdot f^{\prime}\)
Step-by-Step Solution
Verified Answer
True
1Step 1: Identify the Problem Statement
The exercise asks us to verify whether the derivative of the product of two functions, specifically \( x \cdot f(x) \), equals \( f(x) + x \cdot f'(x) \). We need to determine if this statement is true or false.
2Step 2: Recall the Product Rule
The product rule in calculus states that the derivative of a product of two functions, \( u(x) \) and \( v(x) \), is given by: \( \frac{d}{dx} [u \cdot v] = u'v + uv' \). We will apply this rule to the functions \( x \) and \( f(x) \).
3Step 3: Apply the Product Rule
Let \( u = x \) and \( v = f(x) \). The derivatives are \( u' = \frac{d}{dx}(x) = 1 \) and \( v' = \frac{d}{dx}(f) = f'(x) \). According to the product rule, \[ \frac{d}{dx}(x \cdot f(x)) = 1 \cdot f(x) + x \cdot f'(x) = f(x) + x \cdot f'(x). \]
4Step 4: Compare the Results
The calculated expression \( f(x) + x \cdot f'(x) \) matches the given expression in the problem. Therefore, the derivative calculated using the product rule confirms that the statement is true.
Key Concepts
DifferentiationProduct RuleFunction Derivative
Differentiation
Differentiation is a core concept in calculus that allows us to understand how a function changes at any point. This process involves finding the derivative, which essentially gives us the rate of change or the slope of a function at a specific point.
One of the basic rules to find derivatives is the power rule, where if you have a term like \(x^n\), its derivative is \(nx^{n-1}\).
Differentiation is widely used in various fields such as physics for motion analysis, economics for optimization problems, and engineering for modeling change.
One of the basic rules to find derivatives is the power rule, where if you have a term like \(x^n\), its derivative is \(nx^{n-1}\).
Differentiation is widely used in various fields such as physics for motion analysis, economics for optimization problems, and engineering for modeling change.
- Helps in determining maximum and minimum values of functions.
- Finds slopes of tangents to curves.
- Explores rates of change in real-world contexts.
Product Rule
The product rule is a vital tool in calculus, especially when you need to find the derivative of the product of two functions. It tells us how to handle situations where two functions are multiplied together and want to find the rate of change of that product.
Suppose you have two functions, \(u(x)\) and \(v(x)\). The product rule states that the derivative of their product is: \[\frac{d}{dx}[u \cdot v] = u'v + uv'\] where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\), respectively.
In simple terms, this means you differentiate the first function, leave the second one as it is, and then add the product of leaving the first function as it is and differentiating the second. This rule is particularly useful when dealing with products of polynomials or when functions are intertwined in dependency.
Keep in mind, understanding and applying the product rule correctly is essential for tackling more complex calculus problems involving multiple terms.
Suppose you have two functions, \(u(x)\) and \(v(x)\). The product rule states that the derivative of their product is: \[\frac{d}{dx}[u \cdot v] = u'v + uv'\] where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\), respectively.
In simple terms, this means you differentiate the first function, leave the second one as it is, and then add the product of leaving the first function as it is and differentiating the second. This rule is particularly useful when dealing with products of polynomials or when functions are intertwined in dependency.
Keep in mind, understanding and applying the product rule correctly is essential for tackling more complex calculus problems involving multiple terms.
Function Derivative
A function derivative represents the rate at which the function's value changes as its input changes. It's a fundamental building block of calculus, providing insights into the behavior of functions.
When we talk about the derivative of a function like \(f(x)\), we are looking at how \(f\) responds to slight changes in \(x\).
The notation \(f'(x)\) is commonly used to denote this derivative. Derivatives can tell us if a function is increasing or decreasing and how sharply it rises or falls. This is crucial for predicting future values or understanding past trends.
When we talk about the derivative of a function like \(f(x)\), we are looking at how \(f\) responds to slight changes in \(x\).
The notation \(f'(x)\) is commonly used to denote this derivative. Derivatives can tell us if a function is increasing or decreasing and how sharply it rises or falls. This is crucial for predicting future values or understanding past trends.
- Determines concavity and convexity (shape) of curves.
- Essential for optimization—finding the best or most efficient solution.
- Provides instantaneous rates of change essential for physics and chemistry.
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