Derivative Of Y=4e^(5x^2-2): A Step-by-Step Guide

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Let's dive into finding the derivative of the function y = 4e(5x2 - 2). This is a classic calculus problem that combines the chain rule with the derivative of exponential functions. Don't worry; we'll break it down step by step so you can follow along easily. Whether you're a student tackling homework or just brushing up on your calculus skills, this guide will help you understand the process thoroughly.

Understanding the Problem

Before we start, let's make sure we understand what we're trying to find. The derivative of a function, often denoted as dy/dx or y', represents the instantaneous rate of change of the function. In simpler terms, it tells us how much y changes for a tiny change in x. For our function, y = 4e(5x2 - 2), we want to find this rate of change.

The function involves an exponential term, which means we'll need to use the chain rule. The chain rule is essential when you're differentiating a composite function—a function within a function. In our case, we have the exponential function e^u where u = 5x^2 - 2. The constant 4 is just a coefficient, which makes things a bit simpler.

So, to recap, we're dealing with:

  • A constant coefficient: 4
  • An exponential function: e^(something)
  • A composite function: e(5x2 - 2)

With this understanding, we can now proceed to apply the necessary calculus rules to find the derivative.

Applying the Chain Rule

The chain rule states that if you have a composite function y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x). In simpler terms, you differentiate the outer function while keeping the inner function intact, and then multiply by the derivative of the inner function.

For our function, y = 4e(5x2 - 2), we can identify the outer function as f(u) = 4e^u and the inner function as g(x) = 5x^2 - 2. Let's find the derivatives of these two functions separately.

Derivative of the Outer Function

The derivative of f(u) = 4e^u with respect to u is simply f'(u) = 4e^u. Remember, the derivative of e^u is just e^u, and the constant 4 remains as a coefficient. So, we have:

f'(u) = 4e^u

Derivative of the Inner Function

Next, we need to find the derivative of the inner function g(x) = 5x^2 - 2 with respect to x. This is a straightforward application of the power rule. The power rule states that if h(x) = ax^n, then h'(x) = nax^(n-1). Applying this to our inner function:

  • The derivative of 5x^2 is 2 * 5x^(2-1) = 10x
  • The derivative of -2 (a constant) is 0

So, g'(x) = 10x.

Now that we have both f'(u) and g'(x), we can apply the chain rule to find the derivative of the entire function.

Combining the Derivatives

Using the chain rule, dy/dx = f'(g(x)) * g'(x), we substitute the derivatives we found earlier:

dy/dx = 4e(5x2 - 2) * 10x

Simplifying this expression, we get:

dy/dx = 40x * e(5x2 - 2)

So, the derivative of y = 4e(5x2 - 2) is 40x * e(5x2 - 2). This is our final answer.

Step-by-Step Summary

Let's recap the steps we took to find the derivative:

  1. Identify the Outer and Inner Functions: Recognize that y = 4e(5x2 - 2) is a composite function with an outer function f(u) = 4e^u and an inner function g(x) = 5x^2 - 2.
  2. Find the Derivative of the Outer Function: Differentiate f(u) = 4e^u with respect to u to get f'(u) = 4e^u.
  3. Find the Derivative of the Inner Function: Differentiate g(x) = 5x^2 - 2 with respect to x to get g'(x) = 10x.
  4. Apply the Chain Rule: Use the chain rule formula dy/dx = f'(g(x)) * g'(x) to combine the derivatives: dy/dx = 4e(5x2 - 2) * 10x.
  5. Simplify: Simplify the expression to get the final derivative: dy/dx = 40x * e(5x2 - 2).

By following these steps, you can confidently find the derivative of similar composite functions involving exponential terms.

Common Mistakes to Avoid

When finding derivatives, especially with the chain rule, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Forgetting the Chain Rule: The most common mistake is forgetting to multiply by the derivative of the inner function. Always remember that when you have a composite function, you need to apply the chain rule.
  • Incorrectly Differentiating the Inner Function: Make sure you correctly differentiate the inner function. Double-check your application of the power rule and any other relevant rules.
  • Not Simplifying the Final Answer: Sometimes, you might find the correct derivative but fail to simplify it. Always simplify your answer as much as possible to make it cleaner and easier to work with.
  • Confusing Constants: Be careful with constants. Remember that the derivative of a constant is zero, but a constant coefficient remains as a coefficient in the derivative.

By keeping these common mistakes in mind, you can improve your accuracy and avoid errors when finding derivatives.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Find the derivative of y = 3e(2x3 + 1).
  2. Find the derivative of y = e^(sin(x))
  3. Find the derivative of y = 7e(x2 - 4x + 3).

Working through these problems will help you practice applying the chain rule and reinforce your understanding of derivatives of exponential functions. Good luck!

Conclusion

Finding the derivative of y = 4e(5x2 - 2) involves a straightforward application of the chain rule. By breaking down the function into its outer and inner components, differentiating each separately, and then combining the results, we can arrive at the derivative: dy/dx = 40x * e(5x2 - 2). Remember to avoid common mistakes and practice with similar problems to master this technique. With a solid understanding of the chain rule and derivatives of exponential functions, you'll be well-equipped to tackle more complex calculus problems. Keep practicing, and you'll become more confident in your calculus skills. And hey, if you ever get stuck, just revisit this guide for a quick refresher!