Problem 19 - Precalculus: Mathematical for Calculus

Question

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=x^{2}+2 x$$

Step-by-Step Solution

Answer

\( f'(a) = 2a + 2 \)

Step 1: 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 \( f \).

Step 2: 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 \( 2x \).

Step 3: Differentiate Each Term

Differentiate each term of \( f(x) \):- The derivative of \( x^2 \) is \( 2x \).- The derivative of \( 2x \) is \( 2 \).Thus, \( f'(x) = 2x + 2 \).

Step 4: 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 \( x = a \).

Key Concepts

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 \( 1 \times 2x^{0} = 2 \).

• 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:

• 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.

Here, the differentiation process transforms \( f(x) = x^2 + 2x \) into \( f'(x) = 2x + 2 \). This derivative tells us how \( f(x) \) changes with different \( x \) values.

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—a 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 \( a \):

• 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.

This process demonstrates how derivatives aren't just theoretical but can provide practical results, like predicting the behavior of systems at particular points in their operation.