Derivative Of G(x) = 4e^x - 3x^2 + 2x - 4

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Hey guys! Let's break down how to find the derivative of the function G(x) = 4e^x - 3x^2 + 2x - 4. Derivatives might seem intimidating at first, but once you understand the basic rules, they become much easier to handle. So, grab your pencils, and let's get started!

Understanding Derivatives

Before diving into the specifics, let's quickly recap what a derivative actually is. In simple terms, the derivative of a function tells you the slope of the function at any given point. It measures how the function's output changes as its input changes. Think of it as the instantaneous rate of change.

Why is this important? Well, derivatives are used everywhere in science, engineering, economics, and more! They help us optimize processes, model physical phenomena, and understand complex systems. For example, derivatives can help you find the maximum profit in a business model or determine the velocity of an object at a specific moment.

The notation for a derivative can vary. If we have a function f(x), its derivative can be written as f'(x), dy/dx, or d/dx [f(x)]. All of these notations mean the same thing: we're finding the derivative of the function f with respect to x.

Basic Differentiation Rules

To find the derivative of G(x), we need to apply some basic differentiation rules. Here are the most important ones we'll use:

  1. Power Rule: If f(x) = x^n, then f'(x) = n * x^(n-1).
  2. Constant Multiple Rule: If f(x) = c * g(x), where c is a constant, then f'(x) = c * g'(x).
  3. Sum/Difference Rule: If f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x). Similarly, if f(x) = u(x) - v(x), then f'(x) = u'(x) - v'(x).
  4. Derivative of e^x: If f(x) = e^x, then f'(x) = e^x. Yes, it's that simple!
  5. Derivative of a Constant: If f(x) = c, where c is a constant, then f'(x) = 0.

With these rules in mind, we're ready to tackle the derivative of G(x).

Finding the Derivative of G(x)

Our function is G(x) = 4e^x - 3x^2 + 2x - 4. We'll find the derivative term by term.

  1. Derivative of 4e^x:

    • Using the Constant Multiple Rule and the Derivative of e^x, we have:
    • d/dx [4e^x] = 4 * d/dx [e^x] = 4 * e^x = 4e^x
  2. Derivative of -3x^2:

    • Using the Constant Multiple Rule and the Power Rule, we have:
    • d/dx [-3x^2] = -3 * d/dx [x^2] = -3 * (2 * x^(2-1)) = -3 * 2x = -6x
  3. Derivative of 2x:

    • Using the Constant Multiple Rule and the Power Rule (x = x^1), we have:
    • d/dx [2x] = 2 * d/dx [x] = 2 * (1 * x^(1-1)) = 2 * 1 = 2
  4. Derivative of -4:

    • Using the Derivative of a Constant Rule, we have:
    • d/dx [-4] = 0

Now, we combine these results using the Sum/Difference Rule:

G'(x) = d/dx [4e^x] - d/dx [3x^2] + d/dx [2x] - d/dx [4] G'(x) = 4e^x - 6x + 2 - 0 G'(x) = 4e^x - 6x + 2

Final Answer

Therefore, the derivative of the function G(x) = 4e^x - 3x^2 + 2x - 4 is G'(x) = 4e^x - 6x + 2. And that's it, guys! We've successfully found the derivative. Wasn't so bad, right?

Practice Problems

To solidify your understanding, try these practice problems:

  1. Find the derivative of f(x) = 5x^3 - 2x + 7.
  2. Find the derivative of g(x) = 2e^x + 4x^2 - 1.
  3. Find the derivative of h(x) = -3x^4 + 6x - 9.

Work through these problems step-by-step, applying the differentiation rules we discussed. Check your answers to make sure you're on the right track. The more you practice, the more comfortable you'll become with derivatives.

Common Mistakes to Avoid

When finding derivatives, it's easy to make common mistakes. Here are a few to watch out for:

  • Forgetting the Power Rule: Remember to subtract 1 from the exponent after multiplying by the original exponent.
  • Ignoring the Constant Multiple: Don't forget to multiply the derivative of a term by its constant coefficient.
  • Incorrectly Applying the Chain Rule: The Chain Rule applies when differentiating composite functions (functions within functions). We didn't need it for this problem, but it's essential to understand for more complex derivatives.
  • Confusing Derivatives with Integrals: Derivatives and integrals are inverse operations, but they're not the same thing. Make sure you know which one you're trying to find.

By being aware of these potential pitfalls, you can avoid making costly errors and improve your accuracy.

Further Exploration

If you're interested in learning more about derivatives, there are tons of resources available online and in textbooks. Here are a few topics you might want to explore:

  • The Chain Rule: For differentiating composite functions.
  • The Product Rule: For differentiating the product of two functions.
  • The Quotient Rule: For differentiating the quotient of two functions.
  • Higher-Order Derivatives: Finding the second, third, or even higher derivatives of a function.
  • Applications of Derivatives: Using derivatives to solve real-world problems in optimization, physics, and other fields.

Keep exploring and expanding your knowledge of calculus. The more you learn, the more powerful your problem-solving skills will become.

Conclusion

Finding the derivative of a function like G(x) = 4e^x - 3x^2 + 2x - 4 might seem tricky at first, but by breaking it down into smaller steps and applying the basic differentiation rules, it becomes manageable. Remember to practice regularly and watch out for common mistakes. With a little effort, you'll become a derivative master in no time! Keep up the great work, and happy calculating, guys!