Calculate Derivatives: A Step-by-Step Guide

by ADMIN 44 views

Hey math enthusiasts! Today, we're diving into the world of derivatives. We'll be solving a classic problem: Given a function f(t), find its derivative f'(t) and then evaluate it at a specific point. Let's get started, guys!

Understanding Derivatives

So, what exactly is a derivative? Well, in simple terms, the derivative of a function tells us the rate at which the function's output changes with respect to its input. Think of it as the function's instantaneous rate of change. It's super useful for understanding how things are changing over time or with respect to some other variable. In our case, the function f(t) depends on the variable t. The derivative f'(t) represents how f(t) changes as t changes. Derivatives are the backbone of calculus and are used everywhere from physics and engineering to economics and computer science.

To find the derivative, we need to apply the rules of differentiation. There are several rules, such as the power rule, the product rule, the quotient rule, and the chain rule. Because our function is a product of two other functions, we'll need to use the product rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function. Before we jump into the math, it's worth stressing the importance of understanding the concepts. Don't just memorize formulas; try to visualize what the derivative represents graphically. The derivative is the slope of the tangent line to the function at any given point. So, when you calculate f'(3), you are finding the slope of the tangent line to f(t) when t equals 3. This is incredibly helpful for analyzing the behavior of functions and solving real-world problems. Let’s get our hands dirty and break down the problem step by step.

Now, let's look at the function: f(t) = (t² + 2t + 6)(3t² + 4). We have two parts here, and we're going to treat them separately at first. First, you've got (t² + 2t + 6), and second, you've got (3t² + 4). Because these two things are multiplied together, we're going to use the product rule to get our answer. The product rule states that if we have a function f(t) = u(t) * v(t), then the derivative f'(t) = u'(t) * v(t) + u(t) * v'(t). Where u'(t) and v'(t) are the derivatives of u(t) and v(t) respectively.

So, what does this mean in practice? It means that we'll first find the derivatives of (t² + 2t + 6) and (3t² + 4) separately, and then we will apply them to the product rule formula, like magic. The whole process is very mechanical, so it is important to understand the process. Also, take care to avoid any calculation errors. That is why it is important to write down the steps of the process to avoid mistakes.

Finding the Derivative f'(t)

Alright, let's get down to business and find that derivative, shall we? Remember our function f(t) = (t² + 2t + 6)(3t² + 4). Let's designate:

  • u(t) = t² + 2t + 6
  • v(t) = 3t² + 4

Now, we'll find the derivatives of u(t) and v(t) separately. For u(t) = t² + 2t + 6, the derivative, u'(t), is found using the power rule. The power rule states that the derivative of tⁿ is nt^(n-1)*. So:

  • The derivative of t² is 2t¹ = 2t.
  • The derivative of 2t¹ is 2.
  • The derivative of the constant 6 is 0.

Therefore, u'(t) = 2t + 2. Now, for v(t) = 3t² + 4:

  • The derivative of 3t² is 6t.
  • The derivative of the constant 4 is 0.

So, v'(t) = 6t. Now that we have u(t), v(t), u'(t), and v'(t), we can plug them into the product rule formula: f'(t) = u'(t) * v(t) + u(t) * v'(t).

  • f'(t) = (2t + 2)(3t² + 4) + (t² + 2t + 6)(6t)

Let’s expand and simplify this, shall we? Multiply out the terms: f'(t) = (6t³ + 8t + 6t² + 8) + (6t³ + 12t² + 36t). Combine like terms: f'(t) = 12t³ + 18t² + 44t + 8.

So there you have it, folks! f'(t) = 12t³ + 18t² + 44t + 8. We've successfully found the derivative of our function using the product rule. Now, let’s move on to the second part of the problem.

This is why we have to use the power rule and the product rule. If you are starting out, do not be afraid to make mistakes, the important part is the understanding of the concepts. Doing more exercises will help you understand better.

Calculating f'(3)

Now that we have f'(t) = 12t³ + 18t² + 44t + 8, we need to find the value of the derivative at t = 3. This means we simply substitute t with 3 in our derivative function. It’s like, when someone asks you what’s the value of something when t is a specific number, you just put the number in the place of t. Easy peasy, right?

So, let’s plug in t = 3 into our equation:

  • f'(3) = 12(3)³ + 18(3)² + 44(3) + 8

Now, let's crunch those numbers. First, we calculate the powers: 3³ = 27 and 3² = 9. So, the equation becomes:

  • f'(3) = 12(27) + 18(9) + 44(3) + 8

Next, we multiply:

  • f'(3) = 324 + 162 + 132 + 8

Finally, we add it all up:

  • f'(3) = 626

Therefore, f'(3) = 626. This means that the instantaneous rate of change of the function f(t) at the point t = 3 is 626. Awesome! We've found the derivative of the function, and we've evaluated it at a specific point. You've successfully navigated through a calculus problem, and that's something to be proud of!

The most common mistake is to do not take care of the details, such as multiplying by the wrong number, or calculating the powers. Always double-check your calculations. Also, make sure that you are using the correct rules. Knowing the basic rules of calculus will make your life easier.

Conclusion: You Did It!

We did it, guys! We've successfully calculated the derivative of a function and evaluated it at a given point. You've now got a solid understanding of how to apply the product rule and find derivatives. Remember that practice is key. The more you work through these problems, the more comfortable you'll become. Keep exploring, keep learning, and keep enjoying the beautiful world of mathematics!

Remember, derivatives are a fundamental concept in calculus and have numerous applications in various fields. Understanding how to find and interpret derivatives opens the door to a deeper understanding of how things change. So keep at it, and you'll find that these concepts become more intuitive over time. Keep practicing, and don't be afraid to ask for help when needed. Math can be fun if you let it! So go forth and conquer those calculus problems!