Calculate The Derivative Of The Function. Then Find The Value Of The Derivative As Specified. F(x)= 8/x+2
Understanding how to differentiate functions is fundamental in calculus, as derivatives measure how a function changes at any given point. In this article, we will focus on a specific function: \( F(x) = \frac{8}{x} + 2 \). Our goal is twofold: first, to find the derivative \( F'(x) \), which tells us the rate of change of the function at any value of \( x \), and second, to determine the value of this derivative for specific \( x \)-values. We will explore the steps involved in differentiation, apply the relevant rules, and interpret the results to deepen our understanding of the function’s behavior.
Understanding the Function \( F(x) = \frac{8}{x} + 2 \)
Breaking Down the Function
The function \( F(x) = \frac{8}{x} + 2 \) is composed of two parts:
- \( \frac{8}{x} \) — a rational function that involves division by \( x \)
- \( + 2 \) — a constant term
When differentiating such a function, we will need to apply specific rules suitable for each component, particularly the power rule and the constant rule.
Step-by-Step Differentiation of \( F(x) \)
Rewriting the Function for Ease of Differentiation
To differentiate \( F(x) = \frac{8}{x} + 2 \), it’s often easier to express the rational term as a power of \( x \):
F(x) = 8x^{-1} + 2
This transformation simplifies the application of differentiation rules.
Applying Differentiation Rules
We will use the following rules:
- Power Rule: If \( f(x) = ax^{n} \), then \( f'(x) = a n x^{n-1} \)
- Constant Rule: The derivative of a constant is zero.
Calculating the Derivative \( F'(x) \)
- Differentiate \( 8x^{-1} \):
- Apply the power rule: \( \frac{d}{dx} [8x^{-1}] = 8 \times (-1) x^{-1 - 1} = -8 x^{-2} \)
- Differentiate \( 2 \):
- Derivative of a constant is zero: \( \frac{d}{dx} [2] = 0 \)
Putting it all together, we obtain:
F'(x) = -8 x^{-2} + 0 = -8 x^{-2}
Rewriting the derivative back into a more familiar form:
F'(x) = -\frac{8}{x^{2}}
Interpreting The Derivative \( F'(x) \)
The derivative \( F'(x) = -\frac{8}{x^{2}} \) provides insights into the behavior of the original function:
- Since \( x^{2} \) is always positive for \( x \neq 0 \), the sign of \( F'(x) \) is negative for all \( x \neq 0 \).
- This indicates that the function \( F(x) \) is decreasing everywhere in its domain (excluding \( x=0 \)).
- The magnitude of the derivative decreases as \( |x| \) increases, implying the rate of change becomes less steep for larger \( |x| \).
Finding The Value Of The Derivative at Specific Points
Selecting Points for Evaluation
To fully understand the behavior at particular points, evaluate \( F'(x) \) at specific \( x \)-values:
- \( x = 1 \)
- \( x = -1 \)
- \( x = 2 \)
- \( x = -3 \)
Calculations for Each Point
At \( x = 1 \)
F'(1) = -8 / (1)^2 = -8 / 1 = -8
The derivative is \(-8\), indicating the function decreases at a rate of 8 units per unit increase in \( x \) at \( x=1 \).
At \( x = -1 \)
F'(-1) = -8 / (-1)^2 = -8 / 1 = -8
The derivative is again \(-8\); the function decreases at the same rate at \( x=-1 \) as at \( x=1 \).
At \( x = 2 \)
F'(2) = -8 / (2)^2 = -8 / 4 = -2
The rate of decrease is less steep compared to at \( x=1 \), with the derivative value \(-2\).
At \( x = -3 \)
F'(-3) = -8 / (-3)^2 = -8 / 9 \approx -0.8889
The derivative is approximately \(-0.8889\), indicating a gentle decline at this point.
Summary of Results and Their Significance
The derivative \( F'(x) = -\frac{8}{x^{2}} \) reveals several key features of the function:
- Always Negative (except at \( x=0 \)): The function is decreasing for all \( x \neq 0 \).
- The rate diminishes as \( |x| \) increases: The magnitude of the derivative gets smaller with larger \( |x| \), implying the function flattens out at distant points.
- Vertical asymptote at \( x=0 \): Since the derivative involves \( x^{2} \) in the denominator, the derivative is undefined at \( x=0 \), corresponding to the original function’s vertical asymptote.
Conclusion
In this comprehensive exploration, we differentiated the function \( F(x) = \frac{8}{x} + 2 \), resulting in the derivative \( F'(x) = -\frac{8}{x^{2}} \). This derivative provides essential insights into the function’s behavior, showing that it is decreasing everywhere except at \( x=0 \) where it is undefined. Evaluating the derivative at specific points reveals how the rate of change varies across the domain, emphasizing the importance of derivatives in understanding the dynamics of mathematical functions. Mastery of these differentiation techniques equips students and professionals alike to analyze a wide array of functions and their applications in science, engineering, and beyond.
