Problem 26 - Precalculus: Mathematics for Calculus

Question

Find the derivative of the function at the given number. $$G(x)=1+2 \sqrt{x}, \quad \text { at } 4$$

Step-by-Step Solution

Answer

The derivative at 4 is \( \frac{1}{2} \).

Step 1: Identify the function

The function given is \( G(x) = 1 + 2\sqrt{x} \). We want to find the derivative and then evaluate it at a specific point, \( x = 4 \).

Step 2: Use the power rule for derivatives

Rewrite the square root as a power: \( G(x) = 1 + 2x^{1/2} \). Now, differentiate using the power rule: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

Step 3: Differentiate the function

Differentiate each term individually. The derivative of \( 1 \) is \( 0 \) because it's a constant. The derivative of \( 2x^{1/2} \) is \( 2 \times \frac{1}{2}x^{-1/2} = x^{-1/2} \). Therefore, \( G'(x) = x^{-1/2} \).

Step 4: Simplify the derivative

Simplify \( G'(x) \). Since \( x^{-1/2} = \frac{1}{\sqrt{x}} \), the derivative becomes \( G'(x) = \frac{1}{\sqrt{x}} \).

Step 5: Evaluate the derivative at \( x = 4 \)

Substitute \( x = 4 \) into the simplified derivative: \( G'(4) = \frac{1}{\sqrt{4}} \). Calculate \( \sqrt{4} = 2 \), so \( G'(4) = \frac{1}{2} \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus, especially important for differentiation. It states that if you have a function of the form \( f(x) = x^n \), the derivative, denoted \( f'(x) \), is obtained by multiplying \( n \) with \( x \) raised to the power of \( n-1 \). This can be expressed mathematically as \( f'(x) = n x^{n-1} \). The power rule simplifies the process of finding derivatives. Instead of using the definition of the derivative, which involves limits, you can directly apply the power rule to get the result much faster. For example, if \( f(x) = x^3 \), its derivative \( f'(x) \) by using the power rule would be \( 3x^2 \). The power rule is most useful when dealing with polynomial expressions, but it is also adaptable for functions involving fractional exponents, such as \( x^{1/2} \), where you apply the same principle. This flexibility makes it a cornerstone of differentiation techniques.

When applying power rule, remember to:

• Identify the exponent \( n \)
• Multiply the term by \( n \)
• Reduce the exponent by one

Differentiation

Differentiation is the process of finding a derivative, which represents the rate of change of a function with respect to a variable. Importantly, it allows us to determine how a function behaves as its input changes.When you differentiate a function, what you are doing is essentially calculating its slope at any given point. In simpler terms, you are finding whether the function is increasing, decreasing, or neither, at that specific input.To perform differentiation effectively, various rules such as the power rule, product rule, and chain rule, among others, can be used. This exercise focused on using the power rule to differentiate a function that includes a square root, rewritten as a power of \( x \) for easier handling.

Steps often involved in differentiation include:

• Identifying functions and their respective parts or terms
• Applying the appropriate differentiation rules
• Simplifying the resulting expression

Differentiation is not only limited to algebraic expressions. It is equally applicable to find derivatives of trigonometric, exponential, and logarithmic functions, being a foundational aspect of calculus.

Mathematical Functions

Mathematical functions are expressions that take an input and produce an output. Functions are a fundamental part of calculus as they model the relationship between two variables.In the given exercise, the function \( G(x) = 1 + 2\sqrt{x} \) is a simple example of how a mathematical function can be structured. It shows a combination of constant terms and variable components.Understanding functions is crucial as they:

• Allow us to represent real-world phenomena
• Can be graphed to visually interpret data
• Are the basis for applying calculus concepts like differentiation and integration

Function notation is important because it tells us what the function does and what variable it affects. For instance, in \( G(x) \), the function \( G \) acts on the variable \( x \). Knowing how to manipulate and evaluate functions is essential, especially when you need to apply calculus techniques like finding derivatives or solving equations.Functions can come in various forms: polynomial, rational, trigonometric, logarithmic, etc. Each type has its characteristics and rules for differentiation and integration. Understanding these forms helps predict and analyze their behavior when graphed or when applied in different calculations.