Differential Of F(x) = (x^2 Tan X) / (x+1): A Step-by-Step Guide

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Hey guys! Ever stumbled upon a function that looks like a mathematical monster, especially when you need to find its differential? Well, you're not alone! Let's break down one such beast: f(x) = (x^2 * tan x) / (x+1). This might seem intimidating at first, but trust me, with the right tools and a little bit of patience, we can conquer it together. So, let's dive in and make calculus a bit less scary, shall we?

Understanding the Challenge

Before we jump into the solution, it’s good to understand why this particular function might seem tricky. Our function, f(x) = (x^2 * tan x) / (x+1), is a combination of several different types of functions: a polynomial (x^2), a trigonometric function (tan x), and a rational function (division by x+1). To find its differential, we'll need to use a couple of key calculus rules: the product rule and the quotient rule. These rules help us differentiate functions that are formed by multiplying or dividing other functions. Mastering these rules is crucial, not just for this problem, but for a whole bunch of calculus challenges you'll encounter. Think of them as your superhero toolkit for tackling complex derivatives.

The Product Rule: Multiplying Functions

The product rule comes into play when you're differentiating a function that's the product of two other functions. If we have a function like h(x) = u(x) * v(x), then the derivative h'(x) is given by:

h'(x) = u'(x) * v(x) + u(x) * v'(x)

In plain English, you take the derivative of the first function, multiply it by the second function, then add that to the first function multiplied by the derivative of the second function. It's like a little dance between the two functions, each taking turns to be differentiated.

The Quotient Rule: Dividing Functions

The quotient rule is what we use when we're differentiating a function that's the quotient (division) of two other functions. If we have a function like h(x) = u(x) / v(x), then the derivative h'(x) is given by:

h'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2

This one looks a bit more complex, but it's just a systematic way of handling division. You take the derivative of the numerator, multiply it by the denominator, subtract the numerator multiplied by the derivative of the denominator, and then divide the whole thing by the square of the denominator. Think of it as a carefully choreographed routine to keep everything in order.

Step-by-Step Solution: Let's Crack This!

Okay, enough with the theory! Let's get our hands dirty and apply these rules to our function, f(x) = (x^2 * tan x) / (x+1). Remember, the key is to break it down into smaller, manageable parts.

1. Identify the 'Parts'

First, we need to see our function in terms of the quotient rule. Let's identify the numerator and the denominator:

  • u(x) = x^2 * tan x (the numerator)
  • v(x) = x + 1 (the denominator)

Now we've got our 'u' and 'v', ready for the quotient rule treatment.

2. Differentiate u(x): Product Rule in Action

But wait! To differentiate u(x), we need to use the product rule because u(x) itself is a product of two functions: x^2 and tan x. So, let's break it down further:

  • Let p(x) = x^2
  • Let q(x) = tan x

Now we need the derivatives of p(x) and q(x):

  • p'(x) = 2x (using the power rule)
  • q'(x) = sec^2 x (the derivative of tan x)

Time to apply the product rule:

u'(x) = p'(x) * q(x) + p(x) * q'(x) u'(x) = (2x * tan x) + (x^2 * sec^2 x)

Whew! That's the derivative of our numerator. We're making progress!

3. Differentiate v(x): A Simple One

Now for the easier part: differentiating v(x) = x + 1.

  • v'(x) = 1 (the derivative of x is 1, and the derivative of a constant is 0)

Nice and straightforward! One less thing to worry about.

4. Apply the Quotient Rule: Putting It All Together

Now we have all the pieces we need to apply the quotient rule:

f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2

Let's plug in what we've found:

f'(x) = [((2x * tan x) + (x^2 * sec^2 x)) * (x + 1) - (x^2 * tan x) * 1] / (x + 1)^2

5. Simplify (Optional, but Recommended)

Okay, we've got the derivative, but it looks a bit messy. Let's try to simplify it. This step is optional, but it often makes the answer cleaner and easier to work with.

First, let's expand the numerator:

(2x * tan x * (x + 1)) + (x^2 * sec^2 x * (x + 1)) - (x^2 * tan x)

Which expands to:

(2x^2 * tan x + 2x * tan x) + (x^3 * sec^2 x + x^2 * sec^2 x) - x^2 * tan x

Now, let's combine like terms:

x^2 * tan x + 2x * tan x + x^3 * sec^2 x + x^2 * sec^2 x

So, our derivative becomes:

f'(x) = [x^2 * tan x + 2x * tan x + x^3 * sec^2 x + x^2 * sec^2 x] / (x + 1)^2

We can even factor out an x:

f'(x) = [x(x * tan x + 2 * tan x + x^2 * sec^2 x + x * sec^2 x)] / (x + 1)^2

And that, my friends, is our final simplified derivative! 🎉

Key Takeaways: Mastering the Calculus Moves

So, what did we learn from this mathematical adventure? Here are the key takeaways:

  • Break it Down: Complex functions can be tackled by breaking them down into smaller, more manageable parts. Identify the 'u' and 'v' for the product and quotient rules.
  • Know Your Rules: The product and quotient rules are your best friends when differentiating products and quotients of functions. Memorize them, understand them, and use them!
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with these rules. Try different examples and challenge yourself.
  • Simplify: Simplifying your answer not only makes it look nicer but can also make it easier to work with in future calculations.

Why This Matters: Real-World Applications

Okay, so you might be thinking, “When am I ever going to use this in real life?” Well, calculus, including differentiation, has tons of applications in various fields. For example:

  • Physics: Calculating velocity and acceleration.
  • Engineering: Designing structures and optimizing processes.
  • Economics: Modeling economic growth and predicting market trends.
  • Computer Graphics: Creating realistic animations and simulations.

So, the skills you're learning here aren't just abstract mathematical concepts; they're powerful tools that can help you solve real-world problems. Pretty cool, right?

Practice Time: Your Turn to Shine!

Now that we've tackled this beast of a function together, it's your turn to put your skills to the test! Try differentiating similar functions, and don't be afraid to make mistakes. That's how we learn! Remember, calculus is a journey, not a destination. Enjoy the ride, and keep exploring!

I hope this step-by-step guide has helped you understand how to find the differential of f(x) = (x^2 * tan x) / (x+1). Keep practicing, and you'll become a calculus pro in no time! You got this!