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Magazine Chapter 12: Problem 19
Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=x^{2}+2 x$$
Short Answer
Expert verified
\( f'(a) = 2a + 2 \)
Step by step solution
01
Understand the Function
The function given is \( f(x) = x^2 + 2x \). We are required to find its derivative, \( f'(x) \), and then evaluate \( f'(a) \) for a specific \( a \) in the domain of \( f \).
02
Apply the Power Rule
To differentiate \( f(x) = x^2 + 2x \), we will use the power rule which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \). Apply this to \( x^2 \) and \( 2x \).
03
Differentiate Each Term
Differentiate each term of \( f(x) \):- The derivative of \( x^2 \) is \( 2x \).- The derivative of \( 2x \) is \( 2 \).Thus, \( f'(x) = 2x + 2 \).
04
Evaluate the Derivative at \( a \)
Now, evaluate the derivative \( f'(x) \) at \( x = a \). Substitute \( a \) into \( f'(x) \):\[ f'(a) = 2a + 2 \]. This gives the value of the derivative at \( x = a \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule
The power rule is a fundamental technique in calculus, making it easier to differentiate functions that are made up of power terms. It says that if you have a term in the form of \( x^n \), then its derivative is \( nx^{n-1} \). This rule simplifies the process of differentiation significantly.
Consider the function \( f(x) = x^2 + 2x \). The term \( x^2 \) is a power term where \( n = 2 \), and using the power rule, its derivative is \( 2x^{1} = 2x \). For the term \( 2x \), we essentially have \( 2x^1 \). Applying the power rule gives us \( 1 \times 2x^{0} = 2 \).
Consider the function \( f(x) = x^2 + 2x \). The term \( x^2 \) is a power term where \( n = 2 \), and using the power rule, its derivative is \( 2x^{1} = 2x \). For the term \( 2x \), we essentially have \( 2x^1 \). Applying the power rule gives us \( 1 \times 2x^{0} = 2 \).
- Quick and useful: The power rule streamlines the process, saving time.
- Widely applicable: You can use it on any power of \( x \).
- Foundational: It is key to understanding more advanced derivatives.
Differentiation
Differentiation is the process of finding the derivative of a function. A derivative is a measure of how a function changes as its input changes. In simpler terms, it describes the rate of change of a function with respect to one of its variables. Differentiation makes it possible to find the slope of a function at any given point, which is crucial in fields such as physics and economics.
The step-by-step solution involves differentiating a simple polynomial function. For example:
The step-by-step solution involves differentiating a simple polynomial function. For example:
- Start by writing down the function \( f(x) = x^2 + 2x \).
- Differentiate each component using rules like the power rule.
- Combine the derivatives of all terms to find the derivative of the whole function.
Functions
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs. Functions describe how one quantity changes with another, and they are essential components of calculus.
For the given function \( f(x) = x^2 + 2x \), understanding its parts helps in differentiating. This function is an example of a polynomial function鈥攁 function made up of terms involving powers of \( x \). Each polynomial term can be easily differentiated using basic rules, like the power rule.
Some key points about functions:
For the given function \( f(x) = x^2 + 2x \), understanding its parts helps in differentiating. This function is an example of a polynomial function鈥攁 function made up of terms involving powers of \( x \). Each polynomial term can be easily differentiated using basic rules, like the power rule.
Some key points about functions:
- They give structure: Functions provide a way to express relationships mathematically.
- They vary in form: From simple lines to complex curves, functions can take many forms.
- They are differentiable: Most well-behaved functions can be differentiated, meaning we can analyze their rates of change.
Evaluating Derivatives
Once a derivative is found, evaluating it at specific points gives valuable information about the original function's behavior at those points. In the problem, we focus on evaluating \( f'(x) = 2x + 2 \) at \( x = a \).
To evaluate the derivative at \( a \):
To evaluate the derivative at \( a \):
- Substitute \( a \) into the derived formula: \( f'(a) = 2a + 2 \).
- This result shows the rate of change of \( f(x) \) when \( x = a \).
- It provides insights like slope, instantaneous rate of change, etc.
