Unlocking Derivatives: A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of calculus and tackling a classic problem: finding the derivative of y = e^(cosh(6x)). Don't worry if it sounds intimidating at first; we'll break it down into easy-to-digest steps. Derivatives are super important in mathematics, physics, and engineering – they help us understand rates of change, which is crucial for modeling real-world phenomena. So, grab your pencils, and let's get started on this math adventure! We will use the chain rule, one of the most fundamental tools in calculus, which allows us to differentiate composite functions.

Understanding the Chain Rule

Before we jump into the problem, let's quickly review the chain rule. The chain rule is the secret weapon when you're dealing with composite functions – that is, functions within functions. In simple terms, if you have a function y = f(g(x)), the chain rule states that the derivative y' is found by multiplying the derivative of the outer function f'(g(x)) by the derivative of the inner function g'(x). Mathematically, this is expressed as: y' = f'(g(x)) * g'(x). Think of it like peeling an onion: you peel the outer layer (differentiate the outer function) and then work your way inwards, differentiating each layer (inner function) until you're done. This rule is crucial for problems like ours, where we have a function (e^u) where the exponent is another function (cosh(6x)). The chain rule allows us to break down complex derivatives into simpler, manageable steps, and is essential for anyone learning calculus. Mastering the chain rule is akin to unlocking a superpower in the realm of differentiation. It is not just about memorizing a formula; it is about understanding the underlying principle of how rates of change are related across different layers of a composite function. This understanding allows us to solve a variety of intricate problems.

So, let’s begin our journey of discovery and mastery!

Step 1: Identify the Outer and Inner Functions

Okay, let's break down our function: y = e^(cosh(6x)). Here, we have a composite function. The outer function is the exponential function, e^u, and the inner function is cosh(6x). We need to identify these before we apply the chain rule. Recognizing the structure is the first key step. The outer function, e^u, dictates the overall form, while the inner function, cosh(6x), provides the argument for the exponential. Identifying these components correctly sets the stage for accurate differentiation. The ability to correctly identify the components is a crucial step towards mastering calculus. Without this fundamental step, the subsequent application of the chain rule can lead to incorrect results. Therefore, taking the time to precisely identify the outer and inner functions will not only improve the accuracy of our calculations but also solidify our understanding of composite functions. Once we’re comfortable with this, the rest of the problem will be a piece of cake. This careful identification is the bedrock upon which our solution is built.

Specifically:

• Outer function: e^u (where u = cosh(6x))
• Inner function: cosh(6x)

Step 2: Differentiate the Outer Function

Now, let's differentiate the outer function concerning the inner function (which we'll keep in mind). The derivative of e^u with respect to u is simply e^u. This is one of the fundamental properties of the exponential function: its derivative is itself. However, it's important to remember that we're dealing with a composite function, so we must also differentiate the inner function later. For now, we write down the derivative of the outer function, replacing 'u' with the inner function: e^(cosh(6x)).

So, the derivative of the outer function is:

d/du (e^u) = e^u = e^(cosh(6x))

Step 3: Differentiate the Inner Function

Next up, we need to differentiate the inner function, cosh(6x). The derivative of cosh(x) is sinh(x). However, we have cosh(6x), so we need to apply the chain rule again! This time, the inner function is 6x. The derivative of cosh(6x) with respect to x involves differentiating cosh(6x) first, which gives us sinh(6x), and then multiplying by the derivative of 6x with respect to x, which is 6. Therefore, the derivative of the inner function, cosh(6x), is 6 * sinh(6x). The hyperbolic functions, cosh(x) and sinh(x), are essential tools in calculus and have their own unique derivative properties. Grasping the derivatives of hyperbolic functions, and understanding the chain rule, provides us with a versatile approach to solving complex calculus problems. In particular, this step combines understanding the derivative of a standard hyperbolic function with the chain rule. It is a perfect demonstration of how to apply the chain rule to a composite function that contains another composite function. By breaking down cosh(6x) into its components, we can apply our knowledge and solve the problem effectively. This also enhances our ability to handle complex mathematical expressions. So, don't worry, keep going, we're doing great!

So, differentiating the inner function:

d/dx (cosh(6x)) = 6 * sinh(6x)

Step 4: Apply the Chain Rule

Now we're ready to put everything together using the chain rule: y' = f'(g(x)) * g'(x). We have already found both parts: the derivative of the outer function (e^(cosh(6x))) and the derivative of the inner function (6 * sinh(6x)). To find the derivative of the original function y = e^(cosh(6x)), we multiply these two results together. This is where it all comes together! The chain rule helps us combine the individual derivatives. Multiplying these derivatives gives us the final result. At this stage, we have successfully broken down the problem and applied the correct rules of calculus. Now we just need to bring it all together. This final multiplication combines the derivatives of both the outer and inner functions to give us the total derivative of the original function. It shows us how each component interacts to determine the rate of change of the original function. This ability to break down the problem and reconstruct the components to create the final solution truly displays the power of calculus.

Applying the chain rule:

y'(x) = e^(cosh(6x)) * 6 * sinh(6x)

Step 5: Simplify (Optional)

At this point, we have found the derivative! While this is a complete answer, we can often simplify it a little. In this case, we can rearrange the terms to make it look a bit tidier. However, the derivative y'(x) = 6 * sinh(6x) * e^(cosh(6x)) is a perfectly acceptable final answer. Simplification is generally about making the expression more readable or potentially easier to work with in future calculations. In the world of calculus, simplification is a practice of transforming complex expressions into a more concise form without changing their mathematical value. Although we've found the correct answer, rewriting the expression can improve readability and usability. It can also help us discover hidden patterns or properties within the problem. It is worth knowing how to write the solution in different forms. In complex calculus problems, this step is particularly beneficial, as it allows us to handle and manipulate the expression more effectively. While this step is not mandatory, knowing how to do it can greatly improve your ability to handle complex calculus problems.

So, the simplified form is:

y'(x) = 6 * sinh(6x) * e^(cosh(6x))

Conclusion

And there you have it, guys! We have successfully found the derivative of y = e^(cosh(6x)) using the chain rule. This process helps us break down complex derivatives into manageable steps. Remember, the key is to identify the outer and inner functions and apply the chain rule methodically. Keep practicing, and you'll get the hang of it in no time. If you continue practicing, you'll find that this method becomes second nature, and you'll be able to tackle more complex derivatives with confidence. Understanding and applying these concepts lays a strong foundation for future mathematical endeavors. Remember, practice is key, and with enough practice, you’ll become a derivative master! Keep the momentum going, and don't hesitate to ask questions. Keep exploring and experimenting, and soon you'll find yourself able to approach new math challenges with confidence. Well done, guys! You did great!