Derivative Of F(x) = 5x³ + 2x: Calculation And Graphical Meaning

Hey guys! Today, we're diving into a super important concept in calculus: derivatives. Specifically, we're going to figure out the derivative of the function f(x) = 5x³ + 2x and understand what it means graphically. Plus, we'll explore how the derivative relates to the rate of change. So, buckle up and let's get started!

Calculating the Derivative of f(x) = 5x³ + 2x

Okay, so the main question here is: What is the derivative of f(x) = 5x³ + 2x? To find this, we'll use the power rule, which is a fundamental concept in calculus. The power rule states that if you have a term like axⁿ, its derivative is naxⁿ⁻¹. Let's break down how we apply this to our function.

First, let’s look at the function f(x) = 5x³ + 2x. We have two terms here: 5x³ and 2x. We’ll find the derivative of each term separately and then add them together. This is thanks to the sum rule of differentiation, which makes our lives much easier!

For the first term, 5x³, we apply the power rule. Here, a = 5 and n = 3. So, the derivative of 5x³ is 3 * 5x³⁻¹ = 15x². See how we multiplied the coefficient (5) by the exponent (3) and then reduced the exponent by 1? That’s the power rule in action!

Now, let’s tackle the second term, 2x. You can think of 2x as 2x¹. Applying the power rule, we have a = 2 and n = 1. The derivative of 2x is 1 * 2x¹⁻¹ = 2x⁰. Remember that anything to the power of 0 is 1, so 2x⁰ simplifies to 2 * 1 = 2. So, the derivative of 2x is simply 2.

Now, we just add the derivatives of the two terms together. The derivative of 5x³ is 15x², and the derivative of 2x is 2. So, the derivative of f(x) = 5x³ + 2x is 15x² + 2. Awesome, right? We’ve found our answer using the power rule and the sum rule!

So, looking at the options provided: A) 15x² + 2, B) 15x² - 2, C) 5x² + 2, D) 5x³ + 2x², the correct answer is clearly A) 15x² + 2. We nailed it!

Graphical Interpretation of the Derivative

Now that we've calculated the derivative, let's talk about what it actually means graphically. This is where things get really interesting! The derivative, f'(x), gives us the slope of the tangent line to the original function f(x) at any point x. Think of it like this: if you zoom in super close on the graph of f(x) at a particular point, the derivative tells you how steeply the graph is rising or falling at that exact spot.

Imagine you're looking at a curve on a graph. At any point on that curve, you can draw a straight line that just touches the curve at that point – that's the tangent line. The derivative at that point is the slope of this tangent line. A positive derivative means the function is increasing (going uphill), a negative derivative means the function is decreasing (going downhill), and a derivative of zero means the function has a horizontal tangent (a peak or a valley).

Let's consider our derivative, f'(x) = 15x² + 2. Notice that 15x² is always non-negative (because anything squared is non-negative), and we're adding 2 to it. This means that f'(x) is always positive. Graphically, this tells us that the original function, f(x) = 5x³ + 2x, is always increasing. There are no sections where the graph slopes downwards; it's constantly climbing. Pretty neat, huh?

The graphical interpretation is super powerful because it gives us a visual way to understand what the derivative represents. It’s not just a formula; it’s a measure of the steepness and direction of a function’s graph at any given point. This understanding helps us analyze the behavior of functions and solve a ton of real-world problems.

The Derivative as a Rate of Change

Alright, let’s dive into another crucial aspect of the derivative: its meaning as a rate of change. This is where derivatives really shine in practical applications. The derivative f'(x) represents the instantaneous rate of change of the function f(x) with respect to x. In simpler terms, it tells us how much f(x) is changing at any given moment as x changes.

Think about it this way: If f(x) represents the position of a car at time x, then f'(x) represents the car’s velocity at that time. Velocity is the rate of change of position, and that's exactly what the derivative gives us. Similarly, if f(x) represents the population of a city at time x, then f'(x) represents the rate at which the population is growing (or shrinking) at that time. The derivative is like a speedometer for functions!

For our function, f(x) = 5x³ + 2x, the derivative f'(x) = 15x² + 2 tells us how quickly f(x) is changing for different values of x. For example, if x is small, 15x² will be relatively small, and the rate of change will be close to 2. But as x gets larger, 15x² grows much faster, and the rate of change increases dramatically. This means that f(x) is changing more rapidly when x is large compared to when x is small.

The concept of rate of change is incredibly useful in many fields. In physics, it helps us understand motion and acceleration. In economics, it helps us analyze growth rates and marginal costs. In engineering, it’s crucial for designing systems that respond to changes in their environment. So, understanding the derivative as a rate of change unlocks a whole new level of problem-solving power!

To summarize, the derivative provides a way to measure sensitivity. In simple words, it tells us how much the output value of a function will change if we make a slight adjustment to its input value.

Real-World Applications

So, now that we know how to calculate the derivative and what it means graphically and as a rate of change, let's think about some real-world applications. This is where calculus really shines and shows its power!

One classic example is in physics, specifically in the study of motion. If you have a function that describes the position of an object over time, the derivative of that function gives you the object's velocity. And if you take the derivative of the velocity function, you get the object's acceleration. This is super useful for things like designing cars, rockets, and even roller coasters!

In economics, derivatives are used to analyze marginal cost and marginal revenue. Marginal cost is the derivative of the total cost function, and it tells you how much it costs to produce one additional unit of a product. Marginal revenue is the derivative of the total revenue function, and it tells you how much revenue you'll get from selling one additional unit. Businesses use these concepts to make decisions about pricing and production levels.

Engineering is another field where derivatives are essential. For example, in civil engineering, derivatives are used to calculate the stress and strain on structures like bridges and buildings. In electrical engineering, they're used to analyze circuits and design control systems. The applications are virtually endless!

Even in computer science, derivatives play a role. Machine learning algorithms often use derivatives to optimize models and find the best fit for data. The derivative helps the algorithm understand how changing the model's parameters will affect its performance, allowing it to fine-tune the model for better results.

These are just a few examples, but they show how versatile and powerful the concept of the derivative is. Whether you're analyzing the stock market, designing a new airplane, or building a machine learning model, derivatives are a fundamental tool for understanding and predicting change.

Conclusion

Alright guys, that wraps up our deep dive into the derivative of f(x) = 5x³ + 2x! We not only calculated the derivative (15x² + 2) using the power rule, but we also explored its graphical interpretation (the slope of the tangent line) and its meaning as a rate of change (how quickly the function is changing). We even touched on some real-world applications, showing how derivatives are used in physics, economics, engineering, and computer science.

Understanding derivatives is a crucial step in mastering calculus, and it opens the door to a wide range of powerful problem-solving techniques. So, keep practicing, keep exploring, and you'll be amazed at what you can achieve with this awesome tool! Keep up the great work, and I’ll catch you in the next one! Peace out! ✌️