Derivative Of Ln(3t) + 3^(t/2): A Step-by-Step Solution

by Andrew McMorgan 56 views

Hey math enthusiasts! Ever stumbled upon a derivative problem that looks a bit intimidating? Well, today we're going to break down a classic example: finding the derivative of ln(3t) + 3^(t/2). Don't worry, it's not as scary as it looks! We'll go through each step together, making sure you understand the underlying concepts and can tackle similar problems with confidence. So, let's dive in and make some math magic happen!

Breaking Down the Problem

Before we jump into the solution, let's quickly understand what we're dealing with. We have a function that's the sum of two terms: ln(3t) and 3^(t/2). This means we'll be using the sum rule of differentiation, which simply states that the derivative of a sum is the sum of the derivatives. In simpler terms, we can differentiate each term separately and then add the results. This makes the problem much more manageable, guys! We'll also need to remember a couple of key derivative rules: the derivative of the natural logarithm function and the derivative of an exponential function. Keep these in mind, and we'll be golden.

So, our main keyword here is derivative, and we're going to explore how to find it for a specific function. We'll use familiar rules like the sum rule, the chain rule, and the derivatives of logarithmic and exponential functions. By the end of this, you'll not only know the answer but also understand the process behind it. Let's get started!

Step 1: Differentiating ln(3t)

The first part of our problem is finding the derivative of ln(3t). This involves using the chain rule, which is a fundamental concept in calculus. The chain rule comes into play when we have a function within a function, like in this case where we have the natural logarithm function (ln) acting on 3t. Remember, the chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). So, let's break it down:

  • Outer function: f(u) = ln(u)
  • Inner function: g(t) = 3t

First, we need to find the derivatives of these individual functions. The derivative of ln(u) with respect to u is 1/u. This is a standard result that you'll want to memorize. The derivative of 3t with respect to t is simply 3, using the power rule (or just recognizing that the derivative of a constant times t is the constant itself). Now, we apply the chain rule:

d/dt [ln(3t)] = (1 / (3t)) * 3

Simplifying this, we get:

d/dt [ln(3t)] = 1/t

So, the derivative of ln(3t) is 1/t. Not too bad, right? We've successfully navigated the chain rule and found the derivative of our first term. This is a crucial step, and it demonstrates how the chain rule allows us to tackle more complex functions by breaking them down into smaller, more manageable parts. We're one step closer to solving the whole problem! Now, let's move on to the second term and see what we've got.

Step 2: Differentiating 3^(t/2)

Now let's tackle the second term: 3^(t/2). This is an exponential function, and differentiating it requires a slightly different approach. Again, we'll need to use the chain rule, but this time with the derivative of an exponential function. The general rule for the derivative of a^x (where a is a constant) is a^x * ln(a). In our case, we have 3^(t/2), so the base is 3 and the exponent is t/2. Let's break it down step-by-step:

  • Outer function: f(u) = 3^u
  • Inner function: g(t) = t/2

First, we find the derivatives of the individual functions. The derivative of 3^u with respect to u is 3^u * ln(3). This comes directly from the general rule for differentiating exponential functions. The derivative of t/2 with respect to t is 1/2. This is a simple application of the power rule or just recognizing that the derivative of a constant times t is the constant.

Now, we apply the chain rule:

d/dt [3^(t/2)] = 3^(t/2) * ln(3) * (1/2)

We can rewrite this as:

d/dt [3^(t/2)] = (1/2) * 3^(t/2) * ln(3)

So, the derivative of 3^(t/2) is (1/2) * 3^(t/2) * ln(3). We've successfully differentiated the second term using the chain rule and the rule for differentiating exponential functions. Remember, guys, practice makes perfect! The more you work with these rules, the more comfortable you'll become with applying them. Now, let's put it all together and find the final answer.

Step 3: Combining the Results

We've done the hard work of differentiating each term separately. Now comes the fun part: putting it all together! Remember, we're finding the derivative of ln(3t) + 3^(t/2). We used the sum rule, which means we just add the derivatives of each term. We found that:

  • d/dt [ln(3t)] = 1/t
  • d/dt [3^(t/2)] = (1/2) * 3^(t/2) * ln(3)

So, the final derivative is simply the sum of these two results:

d/dt [ln(3t) + 3^(t/2)] = 1/t + (1/2) * 3^(t/2) * ln(3)

And there you have it! We've successfully found the derivative of ln(3t) + 3^(t/2). This might seem like a complex problem at first, but by breaking it down into smaller steps and applying the right rules, we were able to solve it systematically. Remember, the key to calculus is understanding the underlying principles and practicing consistently. Great job, guys! You've just conquered a challenging derivative problem!

Key Takeaways

Let's recap the key concepts we used to solve this problem. This will help solidify your understanding and make you more confident in tackling similar problems in the future:

  1. Sum Rule: The derivative of a sum is the sum of the derivatives. This allowed us to break the problem into two simpler parts.
  2. Chain Rule: This is crucial for differentiating composite functions (functions within functions). We used it for both ln(3t) and 3^(t/2).
  3. Derivative of ln(x): The derivative of ln(x) is 1/x. This is a fundamental result you should memorize.
  4. Derivative of a^x: The derivative of a^x is a^x * ln(a). This is the rule for differentiating exponential functions.

By mastering these concepts, you'll be well-equipped to handle a wide range of derivative problems. Remember, calculus is like a puzzle, and each rule is a piece that fits into the larger picture. Keep practicing, keep exploring, and you'll become a calculus pro in no time!

Practice Problems

Want to test your understanding? Try these practice problems. Applying what you've learned is the best way to solidify your knowledge. Solving math problems can be tricky, but with the right guidance and resources, you can solve them easily. Here are some problems you can try out:

  1. Find the derivative of ln(5x) + 2^(x/3).
  2. Evaluate d/dt [ln(2t) + 4^(t/4)].
  3. Calculate the derivative of ln(t^2) + 5^(2t).

Work through these problems step-by-step, using the techniques we discussed. Check your answers and see how you did. If you get stuck, review the steps we took in this article. Remember, practice is key! And most importantly, don't be afraid to make mistakes. Mistakes are opportunities to learn and grow.

Conclusion

We've successfully navigated the derivative of ln(3t) + 3^(t/2). We saw how to break down the problem using the sum rule, apply the chain rule for composite functions, and use the specific derivative rules for logarithmic and exponential functions. You've gained valuable insights into calculus and hopefully feel more confident in your ability to tackle derivative problems. Keep practicing, guys, and remember that every problem you solve makes you a stronger mathematician. Until next time, happy differentiating!