Finding F'(3) For F(x) = 5g(x) + 3h(x) A Calculus Exploration
In the realm of calculus, understanding the behavior of functions and their rates of change is paramount. The derivative, a fundamental concept, provides a measure of this rate of change. When dealing with composite functions, which are functions formed by combining other functions, the process of finding the derivative requires careful application of differentiation rules. This article delves into the intricacies of finding the derivative of a composite function at a specific point, using a concrete example to illustrate the steps involved.
The Problem: A Symphony of Functions
Let's consider a scenario where we have two differentiable functions, g(x) and h(x). We are given the values of their derivatives at a particular point, x = 3: g'(3) = 4 and h'(3) = -1. Now, imagine a new function, f(x), constructed as a combination of these functions:
f(x) = 5g(x) + 3h(x) + 2
The challenge lies in determining the derivative of f(x) at the point x = 3, denoted as f'(3). This seemingly simple problem unveils the power of differentiation rules and their application in unraveling the behavior of composite functions.
The Power of Differentiation Rules: A Step-by-Step Solution
To find f'(3), we embark on a journey through the world of differentiation rules, each acting as a guiding principle in our quest. The key rules that come into play here are the constant multiple rule, the sum/difference rule, and the constant rule.
1. The Constant Multiple Rule: Scaling the Rate of Change
The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. In mathematical notation:
d/dx [cf(x)] = c * f'(x)
where c is a constant. This rule allows us to scale the rate of change of a function by a constant factor. Applying this rule to our function f(x), we get:
d/dx [5g(x)] = 5g'(x)
d/dx [3h(x)] = 3h'(x)
2. The Sum/Difference Rule: A Symphony of Rates
The sum/difference rule states that the derivative of a sum or difference of functions is equal to the sum or difference of their derivatives. In mathematical notation:
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
This rule allows us to break down the derivative of a complex function into the sum or difference of the derivatives of its simpler components. Applying this rule to our function f(x), we get:
d/dx [5g(x) + 3h(x)] = 5g'(x) + 3h'(x)
3. The Constant Rule: A Stillness in the Rate
The constant rule states that the derivative of a constant is zero. In mathematical notation:
d/dx [c] = 0
where c is a constant. This rule reflects the fact that a constant value does not change, and therefore its rate of change is zero. Applying this rule to our function f(x), we get:
d/dx [2] = 0
4. Unveiling the Derivative: Combining the Rules
Now, we combine these rules to find the derivative of f(x):
f'(x) = d/dx [5g(x) + 3h(x) + 2]
f'(x) = d/dx [5g(x)] + d/dx [3h(x)] + d/dx [2] (Applying the sum rule)
f'(x) = 5g'(x) + 3h'(x) + 0 (Applying the constant multiple and constant rules)
f'(x) = 5g'(x) + 3h'(x)
5. The Final Act: Evaluating at x = 3
We have now found the general expression for f'(x). To find f'(3), we simply substitute x = 3 into the expression:
f'(3) = 5g'(3) + 3h'(3)
We are given that g'(3) = 4 and h'(3) = -1. Substituting these values, we get:
f'(3) = 5(4) + 3(-1)
f'(3) = 20 - 3
f'(3) = 17
Therefore, the derivative of f(x) at x = 3 is 17.
Visualizing the Derivative: A Geometric Interpretation
The derivative, f'(3) = 17, has a geometric interpretation. It represents the slope of the tangent line to the graph of f(x) at the point where x = 3. In other words, it tells us how steeply the function is changing at that particular point. A positive derivative indicates that the function is increasing, while a negative derivative indicates that the function is decreasing. In our case, f'(3) = 17 signifies that the function f(x) is increasing steeply at x = 3.
Applications in the Real World: The Language of Change
The concept of derivatives extends far beyond the realm of pure mathematics. It serves as a fundamental tool in various fields, including physics, engineering, economics, and computer science. Derivatives allow us to model and analyze rates of change in real-world phenomena.
In physics, derivatives are used to describe velocity and acceleration. The velocity of an object is the derivative of its position with respect to time, while acceleration is the derivative of velocity with respect to time. These concepts are crucial for understanding motion and dynamics.
In engineering, derivatives are used in optimization problems. For example, engineers might use derivatives to find the dimensions of a bridge that minimize the amount of material used while still meeting safety requirements.
In economics, derivatives are used to model marginal cost and marginal revenue. Marginal cost is the derivative of the total cost function, while marginal revenue is the derivative of the total revenue function. These concepts are essential for understanding profit maximization.
In computer science, derivatives are used in machine learning algorithms. For example, gradient descent, a widely used optimization algorithm, relies on derivatives to find the minimum of a cost function.
Conclusion: Mastering the Art of Differentiation
Finding the derivative of a composite function at a specific point is a fundamental skill in calculus. By mastering the differentiation rules and understanding their applications, we gain the ability to analyze the behavior of functions and their rates of change. This knowledge empowers us to solve a wide range of problems in mathematics and various other fields. The example we explored in this article serves as a stepping stone to understanding more complex applications of derivatives, paving the way for deeper insights into the world of change and its mathematical representation.
As we have seen, the derivative is not merely a mathematical abstraction; it is a powerful tool that allows us to understand and model the world around us. From the motion of objects to the fluctuations of financial markets, derivatives provide a lens through which we can analyze and interpret the dynamics of change. By embracing the principles of differentiation, we unlock a deeper understanding of the intricate patterns that govern our universe.
