Differentiating E^x+sinx/1+logx: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of calculus to tackle a classic problem: differentiating the function (E^x + sin(x)) / (1 + log(x)). Don't worry, even if you're just starting out, we'll break it down into manageable steps. This guide is designed to be super clear, helping you understand each stage of the differentiation process. We'll be using the quotient rule, a fundamental concept in calculus. So, grab your pencils (or your favorite digital pen!), and let's get started. By the end of this, you'll not only have the answer but also a solid grasp of the underlying principles, so you'll be well-equipped to handle similar problems. We'll explore the differentiation of this composite function, showing each individual step to help you understand the process. The quotient rule can seem a bit intimidating at first, but with a bit of practice, you will master it. The function consists of a combination of exponential, trigonometric, and logarithmic functions, so this example covers a range of common functions in a calculus course. Let's make this fun, shall we?

Understanding the Basics: What is Differentiation?

Before we jump into the problem, let's quickly recap what differentiation is all about. In simple terms, differentiation is a mathematical process used to find the rate at which a function changes. Think of it like this: if you have a graph, differentiation helps you find the slope of the tangent line at any point on that graph. This slope represents the instantaneous rate of change. The primary aim of differentiation is to determine the derivative of a function. The derivative tells us how much the output of the function changes when you make a tiny change to the input. We use differentiation to study motion, optimize designs, and solve all sorts of real-world problems. The derivative is often written as f'(x) or dy/dx, and it represents the rate of change of the function concerning its input variable. So, when we differentiate (E^x + sin(x)) / (1 + log(x)), we're trying to find how this entire function changes as 'x' changes. Understanding this basic concept is key to following along with the rest of the steps. Calculus can seem hard, but we'll show you how to break it down. We'll focus on making sure you're comfortable with the core ideas, which include derivatives and how they work.

The Quotient Rule: Your Differentiation Toolkit

For our specific problem, we're dealing with a fraction, which means we need to use the quotient rule. This is a special formula designed to differentiate functions that are in the form of a fraction (one function divided by another). The quotient rule states that if you have a function f(x) = u(x) / v(x), then its derivative, f'(x), is given by: f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. Where u'(x) is the derivative of the numerator and v'(x) is the derivative of the denominator. Sounds like a mouthful, right? Don't worry, we'll break it down. In our case, u(x) = E^x + sin(x) and v(x) = 1 + log(x). We will find u'(x) and v'(x) separately and then plug them into the quotient rule formula. This will give us the final answer. The quotient rule is a critical skill for any calculus student, so understanding how it works and being able to apply it is super important. The trick to remembering it is to practice it. Make sure you're clear on which part is which, and you'll be fine.

Step-by-Step Differentiation of (E^x + sin(x)) / (1 + log(x))

Alright, guys, let's get down to the actual differentiation. We'll methodically go through each step to make sure you're following along. It’s like a recipe; if you follow the instructions, you'll get the right result. First, let's identify our u(x) and v(x) as we discussed earlier:

  • u(x) = E^x + sin(x)
  • v(x) = 1 + log(x)

Next, we need to find the derivatives of u(x) and v(x). Remember, the derivative of E^x is E^x, and the derivative of sin(x) is cos(x). The derivative of log(x) (natural logarithm) is 1/x. So let's calculate u'(x) and v'(x):

  • u'(x) = E^x + cos(x) (because the derivative of E^x is E^x and the derivative of sin(x) is cos(x))
  • v'(x) = 1/x (because the derivative of 1 is 0 and the derivative of log(x) is 1/x)

Now, we have all the pieces we need. Now, we'll plug these values into the quotient rule formula: f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. This gives us:

f'(x) = [(1 + log(x)) * (E^x + cos(x)) - (E^x + sin(x)) * (1/x)] / (1 + log(x))^2

That's it! We've successfully differentiated our function. We have applied the rule, differentiated the individual components, and combined the results to get the overall derivative. The hard work is done. It's a matter of simplification from here on out. Don't worry if it looks complicated. We will go through the next step to simplify it and make it easier to understand.

Simplifying the Derivative

Now, let's simplify the expression we just derived. This involves expanding the terms and combining like terms where possible. It's a bit of algebraic manipulation, but don't worry, we'll take it step by step. Expanding the numerator, we get:

(1 + log(x)) * (E^x + cos(x)) = E^x + cos(x) + E^x * log(x) + log(x) * cos(x)

(E^x + sin(x)) * (1/x) = E^x/x + sin(x)/x

So, our derivative becomes:

f'(x) = [E^x + cos(x) + E^x * log(x) + log(x) * cos(x) - (E^x/x + sin(x)/x)] / (1 + log(x))^2

This is the simplified form of the derivative. While it might look a bit complex, it's the result of applying the quotient rule and simplifying the resulting expression. The goal of this step is to make the derivative as clean and easy to understand as possible. You might not always be able to simplify it perfectly, but try to eliminate any unnecessary terms. It is important to remember that the simplification may depend on the specifics of the situation and the ultimate use of the derivative. This is the final form of the derivative, but you can manipulate it to find particular results. The simplification step can sometimes be the trickiest part, but with practice, it'll become easier.

Conclusion: Mastering Differentiation

Congratulations, guys! You've successfully differentiated (E^x + sin(x)) / (1 + log(x))! We've walked through the process step by step, from understanding the basics of differentiation and the quotient rule to simplifying the final expression. Remember, practice is key. The more you work through these types of problems, the more comfortable you'll become with the concepts and formulas. Don't hesitate to revisit the steps, try similar problems, and reach out if you have any questions. The world of calculus is vast and rewarding, and with each problem you solve, you're building your skills and understanding. Make sure you practice these problems to better master the concepts. Remember to always double-check your work, especially when it comes to the signs and correct application of the rules. Keep practicing, and you'll be a differentiation pro in no time! Keep exploring and having fun with math; it is a subject that truly reveals the universe's beauty and complexity. You can now confidently tackle a wide array of differentiation problems.