Finding The Derivative Of (3x^4 + X^3)x^(-1) Using The First Principle

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Hey guys! Ever wondered how we find the derivative of a function using the first principle? It might sound intimidating, but trust me, it's super cool once you get the hang of it. Today, we're diving into finding the derivative of the function f(x) = (3x^4 + x3)x(-1), where x ā‰  0, using this method. Buckle up, let's get started!

Understanding the First Principle

So, what exactly is this ā€œfirst principleā€ we keep talking about? In calculus, the first principle, also known as the definition of the derivative or the delta method, is a fundamental concept that allows us to calculate the derivative of a function directly from its basic definition. The derivative, in simple terms, represents the instantaneous rate of change of a function at a given point. Think of it as the slope of the tangent line to the function's curve at that point. The first principle provides a way to find this slope without relying on pre-established differentiation rules. Instead, it goes back to the core idea of a limit.

The formula for the derivative f'(x) using the first principle is:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Where:

  • f'(x) is the derivative of the function f(x).
  • lim(h->0) means we're taking the limit as h approaches 0.
  • h represents a small change in x.
  • f(x + h) is the value of the function at x + h.
  • f(x) is the value of the function at x.

In essence, this formula calculates the slope of a secant line between two points on the function's curve, (x, f(x)) and (x + h, f(x + h)), and then finds the limit of this slope as h gets infinitesimally small. This limit gives us the slope of the tangent line, which is the derivative at that point. The first principle might seem a bit abstract at first, but it's the foundation upon which all differentiation rules are built. By understanding this principle, you gain a deeper appreciation for the power and elegance of calculus. This method is particularly useful for understanding the fundamental concept of derivatives and is especially handy when dealing with functions where standard differentiation rules might not be immediately applicable. It allows us to break down the process into smaller, manageable steps, making it easier to grasp the underlying mechanics of differentiation. So, while it might take a bit more time and effort compared to using shortcuts, mastering the first principle provides a solid foundation for more advanced calculus concepts and problem-solving techniques. Now that we've got a handle on what the first principle is all about, let's jump into applying it to our function.

Step-by-Step Solution

Okay, let's break down how to find the derivative of f(x) = (3x^4 + x3)x(-1) using the first principle. We'll go through it step by step, so you can follow along easily.

1. Simplify the Function

First things first, let's simplify our function. This will make the calculations a whole lot easier. We have:

f(x) = (3x^4 + x^3)x^(-1)

We can distribute the x^(-1) term:

f(x) = 3x^4 * x^(-1) + x^3 * x^(-1)

Using the rule of exponents, x^a * x^b = x^(a+b), we get:

f(x) = 3x^(4-1) + x^(3-1)

f(x) = 3x^3 + x^2

Great! Our simplified function is f(x) = 3x^3 + x^2. This is much easier to work with.

2. Find f(x + h)

Next, we need to find f(x + h). This means we replace every x in our simplified function with (x + h):

f(x + h) = 3(x + h)^3 + (x + h)^2

Now, let's expand those terms. Remember, (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 and (a + b)^2 = a^2 + 2ab + b^2. So,

f(x + h) = 3(x^3 + 3x^2h + 3xh^2 + h^3) + (x^2 + 2xh + h^2)

Distribute the 3:

f(x + h) = 3x^3 + 9x^2h + 9xh^2 + 3h^3 + x^2 + 2xh + h^2

3. Apply the First Principle Formula

Now for the main event! We'll plug f(x + h) and f(x) into the first principle formula:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Substitute the expressions we found:

f'(x) = lim(h->0) [(3x^3 + 9x^2h + 9xh^2 + 3h^3 + x^2 + 2xh + h^2) - (3x^3 + x^2)] / h

4. Simplify the Expression

Time to simplify! Notice that some terms cancel out. We have 3x^3 and x^2 in both the positive and negative forms:

f'(x) = lim(h->0) [9x^2h + 9xh^2 + 3h^3 + 2xh + h^2] / h

Now, we can factor out an h from the numerator:

f'(x) = lim(h->0) h(9x^2 + 9xh + 3h^2 + 2x + h) / h

The h in the numerator and denominator cancel each other out:

f'(x) = lim(h->0) [9x^2 + 9xh + 3h^2 + 2x + h]

5. Evaluate the Limit

Finally, we evaluate the limit as h approaches 0. This means we replace every h with 0:

f'(x) = 9x^2 + 9x(0) + 3(0)^2 + 2x + 0

This simplifies to:

f'(x) = 9x^2 + 2x

And there you have it! The derivative of f(x) = (3x^4 + x3)x(-1) using the first principle is f'(x) = 9x^2 + 2x.

Alternative Method: Power Rule

Just to double-check our work and show you a quicker way for future problems, let's find the derivative using the power rule. Remember, the power rule states that if f(x) = ax^n, then f'(x) = nax^(n-1).

Our simplified function is f(x) = 3x^3 + x^2. Applying the power rule to each term:

  • For 3x^3: The derivative is 3 * 3x^(3-1) = 9x^2.
  • For x^2: The derivative is 2 * x^(2-1) = 2x.

Adding these together, we get:

f'(x) = 9x^2 + 2x

Look at that! We got the same answer as when we used the first principle. This confirms our solution and shows you how powerful the power rule can be as a shortcut.

Conclusion

So, guys, we've successfully found the derivative of (3x^4 + x3)x(-1) using the first principle! It might have seemed like a long process, but you've now seen the fundamental definition of a derivative in action. We also verified our result using the power rule, showing how different methods can lead to the same answer. Understanding the first principle is crucial for grasping the core concepts of calculus, even if you eventually use shortcuts like the power rule. Keep practicing, and you'll become a derivative-finding pro in no time!