Derivative Of Y = 4sin(2x^5)ln(8x^2) / 6x: Calculation
Alright guys, let's dive into calculating the derivative of the function: y = 4sin(2x5)ln(8x2) / 6x. This looks like a juicy problem involving the product rule, chain rule, and quotient rule all rolled into one! Buckle up, it's gonna be a fun ride.
Breaking Down the Function
Before we even think about derivatives, let's simplify the function a bit to make our lives easier. We have y = (4sin(2x5)ln(8x2)) / 6x. We can pull out the constant 4/6, which simplifies to 2/3. So, our function becomes:
y = (2/3) * (sin(2x5)ln(8x2)) / x
Now, this looks a little more manageable. We'll use the quotient rule, which states that the derivative of u/v is (v(du/dx) - u(dv/dx)) / v^2, where u = sin(2x5)ln(8x2) and v = x. Let's find du/dx and dv/dx separately.
Finding du/dx
Here, u = sin(2x5)ln(8x2). To find du/dx, we need the product rule, which states that the derivative of uv is u'(v) + v'(u). Let's set a = sin(2x^5) and b = ln(8x^2). Then u = ab, and du/dx = a'(b) + b'(a).
Finding a' (Derivative of sin(2x^5))
For a = sin(2x^5), we need the chain rule. The derivative of sin(x) is cos(x), and the derivative of 2x^5 is 10x^4. So, a' = cos(2x^5) * 10x^4.
a' = 10x4cos(2x5)
Finding b' (Derivative of ln(8x^2))
For b = ln(8x^2), we also need the chain rule. The derivative of ln(x) is 1/x, and the derivative of 8x^2 is 16x. So, b' = (1 / (8x^2)) * 16x = 16x / 8x^2 = 2/x.
b' = 2/x
Putting it Together: du/dx
Now, we can find du/dx using the product rule: du/dx = a'(b) + b'(a) = (10x4cos(2x5))(ln(8x^2)) + (2/x)(sin(2x^5)).
du/dx = 10x4cos(2x5)ln(8x^2) + (2sin(2x^5)) / x
Alright, that was a mouthful. Now we have du/dx. Let's move on to the easier part: dv/dx.
Finding dv/dx
Since v = x, the derivative of v with respect to x is simply 1.
dv/dx = 1
Easy peasy!
Applying the Quotient Rule
Now we have everything we need to apply the quotient rule. Remember, y = (2/3) * (u/v), so dy/dx = (2/3) * (v(du/dx) - u(dv/dx)) / v^2. Let's plug in what we've found.
dy/dx = (2/3) * [x(10x4cos(2x5)ln(8x^2) + (2sin(2x^5)) / x) - (sin(2x5)ln(8x2))(1)] / x^2
Simplifying the Expression
Let's simplify this beast. First, distribute the x in the numerator:
dy/dx = (2/3) * [10x5cos(2x5)ln(8x^2) + 2sin(2x^5) - sin(2x5)ln(8x2)] / x^2
Now, let's factor out sin(2x^5):
dy/dx = (2/3) * [10x5cos(2x5)ln(8x^2) + sin(2x^5)(2 - ln(8x^2))] / x^2
We can rewrite this as:
dy/dx = (2/3x^2) * [10x5cos(2x5)ln(8x^2) + sin(2x^5)(2 - ln(8x^2))]
And that, my friends, is the derivative of the given function! You can further simplify it, but this form is generally acceptable and highlights the key steps.
Final Answer
dy/dx = (2/3x^2) * [10x5cos(2x5)ln(8x^2) + sin(2x^5)(2 - ln(8x^2))]
Key takeaways: This problem demonstrated a complex application of the quotient rule, product rule, and chain rule. The most crucial aspect is breaking down the function into smaller, manageable parts and applying the rules step-by-step. Remember to be meticulous with your algebra to avoid errors. Don't rush! Take your time and double-check each step. Good luck! This derivative also illustrates the importance of simplification. While the initial application of the rules yields a correct derivative, simplifying it reveals the structure and relationships within the expression more clearly. It can also be helpful for further analysis or use in other calculations.
Additional Tips for Derivatives
- Know Your Rules: Make sure you're comfortable with the power rule, product rule, quotient rule, and chain rule. These are the building blocks for most derivative problems.
- Practice Makes Perfect: The more you practice, the better you'll become at recognizing patterns and applying the appropriate rules. Work through a variety of examples, starting with simpler ones and gradually increasing the complexity.
- Simplify Before Differentiating: If possible, simplify the function before taking the derivative. This can save you a lot of time and effort in the long run.
- Break It Down: Complex functions can be intimidating, but breaking them down into smaller parts can make the process more manageable. Identify the different components of the function and differentiate them separately before combining them.
- Check Your Work: After you've found the derivative, take a moment to check your work. Make sure you haven't made any algebraic errors or forgotten any terms. You can also use a calculator or online tool to verify your answer.
Common Mistakes to Avoid
- Forgetting the Chain Rule: The chain rule is often a source of errors, especially when dealing with composite functions. Remember to multiply by the derivative of the inner function.
- Misapplying the Product or Quotient Rule: Make sure you're using the product and quotient rules correctly. Pay attention to the order of the terms and the signs.
- Algebraic Errors: Algebraic errors can easily creep into derivative calculations, especially when dealing with complex expressions. Double-check your work to make sure you haven't made any mistakes.
- Not Simplifying: Failing to simplify the derivative can make it difficult to work with and can also lead to errors in subsequent calculations. Simplify the derivative as much as possible.
By following these tips and avoiding common mistakes, you can improve your ability to calculate derivatives and succeed in calculus.
Conclusion
So, there you have it! We successfully navigated the maze of trigonometric functions, logarithms, and algebraic manipulations to find the derivative of the given function. Remember, the key is to break down the problem into smaller, manageable steps and apply the rules systematically. Don't be afraid to take your time and double-check your work. And most importantly, keep practicing! The more you practice, the more comfortable you'll become with these concepts. Keep up the awesome work, and until next time, happy differentiating! Remember that calculus is a tool and like any tool, it requires practice to master. Understanding the underlying principles and applying them consistently is key to success in calculus and related fields.