Derivative Of E^(3x+2)^2: A Step-by-Step Guide

by Andrew McMorgan 47 views

Hey guys! Today, we're diving into a super interesting calculus problem: finding the derivative of $e{(3x+2)2}$. This might look a bit intimidating at first, but don't worry, we'll break it down step by step so it's easy to follow. Grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's quickly understand what we're dealing with. We have a composite function here, meaning a function within a function. Specifically, we have an exponential function ($e^u$) where the exponent u is another function $(3x+2)^2$. To find the derivative, we'll need to use the chain rule. Remember, the chain rule is your best friend when dealing with composite functions!

The chain rule states that if we have a function y = f(g(x)), then the derivative of y with respect to x is given by: $\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$. In simpler terms, it means we take the derivative of the outer function, keeping the inner function as it is, and then multiply by the derivative of the inner function. This might seem abstract now, but it'll become clearer as we apply it to our problem.

So, in our case, the outer function is $e^u$ and the inner function is $(3x+2)^2$. We'll first find the derivative of $e^u$ with respect to u, and then find the derivative of $(3x+2)^2$ with respect to x. Finally, we'll multiply these two derivatives together to get our final answer. Make sense? Great! Let's move on to the actual calculation.

Step-by-Step Solution

Step 1: Identify the Outer and Inner Functions

As we discussed, our outer function is $f(u) = e^u$ and our inner function is $g(x) = (3x+2)^2$. Identifying these correctly is crucial for applying the chain rule effectively. Always start by peeling away the layers of the function to see what's inside what. Think of it like unwrapping a present – you need to get to the core!

Step 2: Find the Derivative of the Outer Function

The derivative of $e^u$ with respect to u is simply $e^u$. That's one of the beautiful things about the exponential function – it's its own derivative! So we have: $\frac{df}{du} = e^u$. Remember, we're keeping the inner function as it is for now, so we'll substitute u back with $(3x+2)^2$ later.

Step 3: Find the Derivative of the Inner Function

Now, let's find the derivative of $g(x) = (3x+2)^2$ with respect to x. Here, we'll again use the chain rule, but this time on a simpler function. We can think of this as another layer of the problem. Let $v = 3x+2$, so $g(x) = v^2$. Then, $\frac{dg}{dx} = \frac{dg}{dv} \cdot \frac{dv}{dx}$.

The derivative of $v^2$ with respect to v is $2v$, so $\frac{dg}{dv} = 2v$. The derivative of $3x+2$ with respect to x is 3, so $\frac{dv}{dx} = 3$. Therefore, $\frac{dg}{dx} = 2v \cdot 3 = 6v$. Now substitute v back with $3x+2$, so $\frac{dg}{dx} = 6(3x+2) = 18x + 12$.

Step 4: Apply the Chain Rule

Now that we have the derivatives of both the outer and inner functions, we can apply the chain rule: $\fracdy}{dx} = \frac{df}{du} \cdot \frac{dg}{dx}$. We found that $\frac{df}{du} = e^u$ and $\frac{dg}{dx} = 18x + 12$. So, $\frac{dy}{dx} = e^u \cdot (18x + 12)$. Finally, substitute u back with $(3x+2)^2$ to get our final answer $\frac{dy{dx} = e{(3x+2)2} \cdot (18x + 12)$.

Step 5: Simplify (Optional)

We can simplify the expression a bit by factoring out a 6 from the second term: $\frac{dy}{dx} = 6(3x + 2)e{(3x+2)2}$. This is the derivative of $e{(3x+2)2}$.

Alternative Method: Direct Application of the Chain Rule

Alternatively, you can directly apply the chain rule without explicitly defining u and v. This method requires a bit more mental agility, but it can save you some time once you get the hang of it. Here's how it works:

  1. Recognize the composite function: We have $e{(3x+2)2}$.
  2. Differentiate the outer function: The derivative of $e^x$ is $e^x$, so the derivative of $e{(3x+2)2}$ is $e{(3x+2)2}$, keeping the exponent as it is.
  3. Multiply by the derivative of the inner function: The inner function is $(3x+2)^2$. Using the power rule and chain rule, its derivative is $2(3x+2) \cdot 3 = 6(3x+2) = 18x + 12$.
  4. Combine the results: Multiply the derivative of the outer function by the derivative of the inner function: $e{(3x+2)2} \cdot (18x + 12) = 6(3x + 2)e{(3x+2)2}$.

As you can see, both methods lead to the same result. Choose the one that you find more comfortable and easier to understand.

Key Takeaways

  • The chain rule is essential for differentiating composite functions.
  • Identify the outer and inner functions correctly.
  • Remember that the derivative of $e^u$ is $e^u$.
  • Simplify your answer whenever possible.
  • Practice makes perfect! The more you practice, the easier these problems will become. And really understanding the chain rule is important to solving the problem. It's the fundamental to solve a math question like this.

Common Mistakes to Avoid

  • Forgetting the chain rule: This is the most common mistake. Always remember to multiply by the derivative of the inner function.
  • Incorrectly identifying the outer and inner functions: Make sure you know which function is inside which.
  • Making algebraic errors: Be careful with your algebra, especially when simplifying the expression.
  • Not simplifying the final answer: While not always required, simplifying can make the answer cleaner and easier to work with in further calculations.

Practice Problems

To solidify your understanding, try these practice problems:

  1. Find the derivative of $e{(x2 + 1)^3}$.
  2. Find the derivative of $\sin(e^{2x})$.
  3. Find the derivative of $\ln((4x - 1)^2)$.

Work through these problems step by step, and don't hesitate to refer back to this guide if you get stuck. Remember, the key is to break down the problem into smaller, manageable parts.

Conclusion

So there you have it! Finding the derivative of $e{(3x+2)2}$ isn't as scary as it looks. By understanding the chain rule and breaking the problem down into smaller steps, you can conquer even the most complex calculus problems. Keep practicing, and you'll become a derivative master in no time! Keep it classy, Plastik Magazine readers!