Problem 89
Question
Find a formula for \(\frac{d}{d x}[f(x)]^{2}\) by writing it as \(\frac{d}{d x}[f(x) f(x)]\) and using the Product Rule. Be sure to simplify your answer.
Step-by-Step Solution
Verified Answer
The derivative is \(2f(x)f'(x)\).
1Step 1: Understand the Problem
We need to find the derivative of the function \([f(x)]^2\). This can be rewritten as \(f(x) \cdot f(x)\), allowing us to use the Product Rule for differentiation.
2Step 2: Apply the Product Rule
The Product Rule states that the derivative of two functions multiplied together, \(u(x) \cdot v(x)\), is \(u'(x) v(x) + u(x) v'(x)\). Here, both \(u(x)\) and \(v(x)\) are equal to \(f(x)\). Thus, the derivative \(\frac{d}{dx}[f(x) \cdot f(x)]\) becomes \(f'(x)f(x) + f(x)f'(x)\).
3Step 3: Simplify the Expression
The expression \(f'(x)f(x) + f(x)f'(x)\) can be simplified. Both terms are identical and can be combined into a single term: \(2f(x)f'(x)\).
4Step 4: State the Simplified Formula
Thus, the derivative of \([f(x)]^2\) is \(2f(x)f'(x)\).
Key Concepts
Product RuleDerivative FormulaFunction Notation
Product Rule
When you are learning calculus differentiation, one rule that stands out is the "Product Rule." This rule is particularly useful when you have two functions multiplied together, and you need to find the derivative of their product. Here's an easy way to remember it:
The Product Rule states that if you have two functions, let's say \( u(x) \) and \( v(x) \), then the derivative of their product is given by:
This rule is particularly handy because it allows us to break down complex multiplication into manageable bits, making differentiation straightforward. Understanding and applying the Product Rule is crucial in solving real-world applications where multiple changing factors are involved.
The Product Rule states that if you have two functions, let's say \( u(x) \) and \( v(x) \), then the derivative of their product is given by:
- \( (uv)' = u'v + uv' \)
This rule is particularly handy because it allows us to break down complex multiplication into manageable bits, making differentiation straightforward. Understanding and applying the Product Rule is crucial in solving real-world applications where multiple changing factors are involved.
Derivative Formula
A derivative formula represents the rate at which a function is changing at any point. It is an essential tool in calculus that lets you understand how a quantity changes over time. In simple terms, the derivative shows the slope of a tangent line to the curve of a function at any given point.
The fundamental idea is that if you have a function \( f(x) \), its derivative, denoted as \( f'(x) \), tells you how the function behaves. For instance, if \( f(x) \) represents the position of a car over time, \( f'(x) \) would represent its speed.
To differentiate, you apply various rules (like the Product Rule) to manipulate the function into a form where you can find this rate of change. The power of derivatives is their ability to provide insight into the functional behavior instantly—whether it's rising or falling, accelerating or decelerating.
Calculating derivatives becomes second nature once you understand their rules and formulas, leading to more advanced applications like optimization and predicting trends.
The fundamental idea is that if you have a function \( f(x) \), its derivative, denoted as \( f'(x) \), tells you how the function behaves. For instance, if \( f(x) \) represents the position of a car over time, \( f'(x) \) would represent its speed.
To differentiate, you apply various rules (like the Product Rule) to manipulate the function into a form where you can find this rate of change. The power of derivatives is their ability to provide insight into the functional behavior instantly—whether it's rising or falling, accelerating or decelerating.
Calculating derivatives becomes second nature once you understand their rules and formulas, leading to more advanced applications like optimization and predicting trends.
Function Notation
Function notation is a way to represent functions that makes it easy to read and understand complex expressions. In mathematics, it allows us to express functions using symbols and operations.
A function is usually denoted by \( f(x) \), where \( f \) is the function name and \( x \) is the variable. This notation helps in understanding the type of function and the variables involved without describing the entire function every time.
Using function notation makes the presentation cleaner and more organized, especially when working with calculus concepts like derivatives. For instance, when discussing the derivative, you might see it written as \( f'(x) \) to denote the function's rate of change.
Understanding function notation is important because it sets the foundation for tackling more complex calculus problems, making the notation system a language of its own in mathematics.
A function is usually denoted by \( f(x) \), where \( f \) is the function name and \( x \) is the variable. This notation helps in understanding the type of function and the variables involved without describing the entire function every time.
Using function notation makes the presentation cleaner and more organized, especially when working with calculus concepts like derivatives. For instance, when discussing the derivative, you might see it written as \( f'(x) \) to denote the function's rate of change.
Understanding function notation is important because it sets the foundation for tackling more complex calculus problems, making the notation system a language of its own in mathematics.
Other exercises in this chapter
Problem 88
Derive the Quotient Rule from the Product Rule as follows. a. Define the quotient to be a single function, $$ Q(x)=\frac{f(x)}{g(x)} $$ b. Multiply both sides b
View solution Problem 89
True or False: If \(\lim _{x \rightarrow 2^{+}} f(x)=7\), then \(\lim _{x \rightarrow 2} f(x)=7\)
View solution Problem 90
True or False: If \(\lim _{x \rightarrow 2} f(x)=7\), then \(\lim _{x \rightarrow 2^{+}} f(x)=7\)
View solution Problem 90
Find a formula for \(\frac{d}{d x}[f(x)]^{-1}\) by writing it as \(\frac{d}{d x}\left[\frac{1}{f(x)}\right]\) and using the Quotient Rule. Be sure to simplify y
View solution