Derivative Of Y = X^4 * E^x: A Step-by-Step Guide

Hey guys! Today, let's dive into a common calculus problem: finding the derivative of the function y = x^4 * e^x. This is a classic example that combines the power rule and the product rule, so it's a fantastic way to brush up on these essential calculus techniques. We'll break it down step by step, making sure you understand the why behind each move, not just the how. So, grab your pencils, and let's get started!

Understanding the Basics: Power Rule and Product Rule

Before we jump into the main problem, let's quickly recap the two fundamental rules we'll be using. These are the bread and butter of differentiation, so having them down cold is super important.

The Power Rule

The power rule is your best friend when dealing with terms like x raised to a power. It states that if you have a function of the form f(x) = x^n, where n is any real number, then its derivative f'(x) is given by:

f'(x) = n * x^(n-1)

In simpler terms, you bring the exponent down as a coefficient and then subtract 1 from the exponent. Easy peasy, right? For example, if f(x) = x^3, then f'(x) = 3x^2. This rule will be crucial when we deal with the x^4 part of our function.

The Product Rule

The product rule comes into play when you're differentiating a function that is the product of two other functions. If you have a function y = u(x) * v(x), where u(x) and v(x) are both functions of x, then the derivative y' is given by:

y' = u'(x) * v(x) + u(x) * v'(x)

What does this mean in plain English? It means you take the derivative of the first function, multiply it by the second function, then add that to the first function multiplied by the derivative of the second function. Think of it like a criss-cross applesauce motion, taking derivatives and multiplying. In our case, y = x^4 * e^x, so we have one function x^4 multiplied by another function e^x; hence, the product rule is our go-to method. This rule ensures we account for how both parts of the product contribute to the overall rate of change. Understanding and applying the product rule correctly is essential for accurately finding derivatives in many calculus problems.

Applying the Product Rule to y = x^4 * e^x

Okay, now that we've refreshed our memory on the power and product rules, let's tackle our main problem: finding the derivative of y = x^4 * e^x. This is where things get interesting! We're going to break it down step by step so you can see exactly how the product rule works in action.

Step 1: Identify u(x) and v(x)

The first thing we need to do is identify our two functions, u(x) and v(x). Remember, the product rule applies when we have a function that's the product of two other functions. In our case, it's pretty clear:

• u(x) = x^4
• v(x) = e^x

We've got our x^4 term and our exponential e^x term. Now we know what we're working with. Labeling these clearly is a great way to keep things organized and avoid confusion down the road.

Step 2: Find u'(x) and v'(x)

Next up, we need to find the derivatives of both u(x) and v(x). This is where the power rule and our knowledge of exponential functions come into play.

• For u(x) = x^4, we use the power rule. Bring down the 4, subtract 1 from the exponent, and we get:
  • u'(x) = 4x^3
• For v(x) = e^x, this one's a bit special. The derivative of e^x is simply e^x. Yes, it's that straightforward!
  • v'(x) = e^x

So, we've found the derivatives of our two component functions. Make sure you're comfortable with these basic derivative rules before moving on. Getting these right is crucial for the next step.

Step 3: Apply the Product Rule Formula

Now comes the fun part: plugging everything into the product rule formula. Remember, the product rule states:

y' = u'(x) * v(x) + u(x) * v'(x)

We've already identified u(x), v(x), u'(x), and v'(x), so let's substitute them in:

y' = (4x^3) * (e^x) + (x^4) * (e^x)

See how we just slotted in the pieces we found earlier? This is the heart of applying the product rule. Now we have an expression for the derivative, but we're not quite done yet. We can simplify this a bit further.

Simplifying the Derivative

We've found the derivative using the product rule, but it's always a good idea to simplify your answer as much as possible. This not only makes the expression cleaner but can also make it easier to work with in future calculations. In our case, we can factor out some common terms.

Step 4: Factor Out Common Terms

Looking at our derivative expression:

y' = (4x^3) * (e^x) + (x^4) * (e^x)

We can see that both terms have x^3 and e^x as common factors. So, let's factor those out:

y' = x^3 * e^x * (4 + x)

And there you have it! We've simplified the derivative. Factoring out common terms not only cleans up the expression but also reveals the structure of the derivative more clearly. This simplified form is much easier to understand and use in further analysis or applications.

The Final Answer and What It Means

So, after all that work, what's our final answer? The derivative of y = x^4 * e^x is:

y' = x^3 * e^x * (4 + x)

But what does this derivative actually tell us? Well, the derivative represents the instantaneous rate of change of the function y with respect to x. In simpler terms, it tells us how much y is changing at any given point x. This is incredibly useful in various applications, from physics to economics.

Understanding the Rate of Change

The derivative gives us the slope of the tangent line to the curve of the function at any point. So, if we plug in a specific value for x into our derivative, we get the slope of the tangent line at that x-value. This slope indicates whether the function is increasing, decreasing, or staying constant at that point.

For example, if we plug in x = 0 into our derivative, we get y' = 0. This means that at x = 0, the function has a horizontal tangent line, indicating a potential local minimum or maximum.

Applications in the Real World

Derivatives are not just abstract mathematical concepts; they have tons of real-world applications. Here are just a few:

• Physics: Derivatives are used to calculate velocity and acceleration. If you know the position of an object as a function of time, you can take the derivative to find its velocity and take the derivative again to find its acceleration.
• Economics: Derivatives can be used to find marginal cost and marginal revenue. These concepts help businesses make decisions about production levels and pricing.
• Engineering: Derivatives are used in optimization problems, such as finding the optimal shape for a bridge or the most efficient way to design a circuit.

Common Mistakes to Avoid

When working with derivatives, especially when applying rules like the product rule, it's easy to make mistakes. Let's go over some common pitfalls to help you avoid them.

Forgetting the Product Rule

The most common mistake is simply forgetting to use the product rule when it's needed. Remember, if you're differentiating a product of two functions, you must use the product rule. Don't try to take the derivative of each part separately and multiply them; that's a recipe for disaster!

Incorrectly Applying the Power Rule

Another frequent error is misapplying the power rule. Make sure you bring the exponent down as a coefficient and subtract 1 from the exponent. A simple mistake here can throw off your entire answer.

Messing Up the Derivative of e^x

While the derivative of e^x is e^x, which seems straightforward, it's still a point where errors can creep in, especially in more complex problems. Always double-check that you've correctly applied this rule.

Not Simplifying the Answer

While it's not technically a mistake in the differentiation process itself, not simplifying your answer can make further calculations more difficult and hide the true nature of the derivative. Always try to simplify by factoring out common terms or using algebraic identities.

Practice Makes Perfect

Like any skill, mastering derivatives takes practice. The more you work through problems, the more comfortable you'll become with identifying which rules to use and applying them correctly. So, don't be afraid to tackle lots of different examples.

Try These Practice Problems

Here are a few problems you can try on your own to practice using the product rule and other derivative rules:

1. Find the derivative of y = x^2 * sin(x)
2. Find the derivative of y = (x^3 + 1) * cos(x)
3. Find the derivative of y = e^x * ln(x)

Work through these problems step by step, and be sure to check your answers. The key is to break down each problem into smaller parts and apply the rules systematically.

Conclusion

Finding the derivative of y = x^4 * e^x is a great example of how to use the product rule in calculus. We've walked through the process step by step, from identifying the functions u(x) and v(x) to applying the product rule formula and simplifying the result. Remember, the key to mastering calculus is understanding the underlying principles and practicing regularly. So, keep practicing, and you'll become a derivative pro in no time! If you have any questions, don't hesitate to ask. Happy differentiating, guys!