Problem 11 - Single Variable Calculus

Question

Find the derivative of the function. \(f(x)=x^{2}-2 x+8\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=x^{2}-2 x+8\) is \(f'(x) = 2x - 2\).

Step 1: Identify the terms of the function

In our given function, we have three terms: \(x^2\), \(-2x\), and \(8\). We will apply the power rule to each term separately.

Step 2: Apply the power rule to each term

Applying the power rule to each term: 1. For the term \(x^2\): Using the power rule, \(n=2\). So, we get \(f'(x) = 2 \cdot x^{2-1} = 2x^1 = 2x\). 2. For the term \(-2x\): Using the power rule, \(n=1\). So, we get \(f'(x) = 1 \cdot (-2) \cdot x^{1-1} = -2x^0 = -2\). 3. For the constant term \(8\): Since it's a constant term, its derivative is \(0\).

Step 3: Combine the derivatives

Now, we'll combine the derivatives of each term to get the derivative of the whole function: \(f'(x) = 2x - 2\)

Step 4: Final answer

The derivative of the function \(f(x)=x^2-2x+8\) is: \(f'(x) = 2x - 2\)

Key Concepts

Power Rule

The power rule is an essential technique in calculus used to find the derivative of a function that includes polynomial terms. It's generally stated as follows: if you have a term in the form of \( x^n \), the derivative with respect to \( x \) is \( n \cdot x^{n-1} \). This simple rule allows us to systematically differentiate each term of a polynomial.
For example:

• If you have \( x^4 \), the derivative is \( 4x^3 \).
• For a term like \( 5x^3 \), it becomes \( 15x^2 \).

To use the power rule correctly:

• Identify the power \( n \) of each term.
• Multiply by \( n \) and decrease the power by one.

Applying the power rule can greatly simplify the process of differentiation, especially when dealing with complex polynomials.

Differentiation

Differentiation is a fundamental tool in calculus that provides a way to derive a function's derivative, representing the rate of change of the function with respect to a variable. In essence, differentiation helps determine how the output of a function changes as its input changes.
When differentiating a polynomial function:

• Apply the power rule to each polynomial term.
• Sum up the results to get the derivative of the entire function.
• Remember that constants become zero because they do not change.

For instance, differentiating \( f(x) = x^2 - 2x + 8 \) involves:

• Using the power rule on \( x^2 \) to get \( 2x \).
• Applying it on \( -2x \) to derive \( -2 \).
• The constant \( 8 \) turns into 0 because its derivative is zero.

Completing these steps, you find the derivative \( f'(x) = 2x - 2 \), indicating how the function behaves as \( x \) changes.

Calculus

Calculus is the branch of mathematics that studies change, represented by two main concepts: derivatives and integrals. Derivatives (like those found through differentiation) focus on rates of change, while integrals accumulate quantities. Together, these concepts are pivotal for understanding and analyzing mathematical models of change.
Derivatives, one of the pillars of calculus, describe how fast something changes. This can be applied to physical phenomena like velocity, or to more abstract concepts like the slope of a curve at a given point.

• In physics, we use derivatives to find velocity and acceleration.
• Economics applies derivatives to maximize profit functions.

Understanding calculus involves mastering techniques like the power rule, which simplifies finding derivatives in polynomial functions. By learning the step-by-step process of differentiation and applying it to various functions, students gain insights into the dynamic nature of mathematical systems around them.