Quickly Check my Work Please|Finding Derivatives

In summary, the derivative of the given function is y'=\frac{8x^4+5x^2+6}{4x^4}. The original expression can also be simplified before taking the derivative, resulting in a different but equivalent form of the answer.
  • #1
Raza
203
0

Homework Statement


[tex]y=\frac{8x^4-5x^2-2}{4x^3}[/tex]

Find y'.

2. The attempt at a solution
[tex]y=\frac{8x^4-5x^2-2}{4x^3}[/tex]

[tex]y'=\frac{(32x^3-10x)(4x^3)-(12x^2)(8x^4-5x^2-2)}{(4x^3)^2}[/tex]

[tex]y'=\frac{128x^6-40x^4-96x^6+60x^4+24x^2}{16x^6}[/tex]

[tex]y'=\frac{32x^6+20x^4+24x^2}{16x^6}[/tex]


[tex]y'=\frac{4x^2(8x^4+5x^2+6)}{4x^2(4x^4)}[/tex]
The 4x2 gets canceled out.

Finally,

[tex]y'=\frac{8x^4+5x^2+6}{4x^4}[/tex]



Did I do it right?
 
Last edited:
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  • #2
Perrrfect.

edit: The second and third terms in the numerator (of the final result) needs some corrections.
 
Last edited:
  • #3
Thank you very much!
 
  • #4
Rather than using the quotient rule, it would be quicker to simplify the original expression...

[tex]y=\frac{8x^4-5x^2-2}{4x^3}[/tex]

[tex]= 2x - \frac{5}{4} x^{-1} - \frac{1}{2}x^{-3}[/tex]

Then, take the derivative. The form of the answer won't be the same as what you have, but it'll be equivalent.
 
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1. How do I quickly check my work for finding derivatives?

To quickly check your work for finding derivatives, you can use the product rule, quotient rule, or chain rule depending on the type of function you are differentiating. You can also use online calculators or check your work with the help of a tutor or classmate.

2. What is the process for finding derivatives?

The process for finding derivatives involves using various rules and formulas, such as the power rule, product rule, quotient rule, and chain rule. It also involves understanding the relationship between the original function and its derivative.

3. How can I be sure that I have correctly found the derivative?

To ensure that you have correctly found the derivative, you can compare your answer to the answer key or use online calculators to confirm your work. You can also check your work by taking the derivative of your answer and seeing if it matches the original function.

4. What are some common mistakes to avoid when finding derivatives?

Some common mistakes to avoid when finding derivatives include forgetting to apply the correct rule, making algebraic errors, and forgetting to simplify your final answer. It is also important to double-check your work for any sign errors or incorrect use of the chain rule.

5. Can I use shortcuts to find derivatives more quickly?

While there may be some shortcuts or tricks to finding derivatives, it is important to understand the underlying principles and rules in order to accurately find derivatives. It is not recommended to solely rely on shortcuts as they may lead to incorrect answers.

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