How Does Implicit Differentiation Apply to Polynomial Equations?

[recoil33]

Question:

y⁶ + 6 (x^2+4)⁶ = 9

6y⁵ .dy/dx . 6(x² + 4)⁵ . (2x) = 0

6y⁵ .dy/dx = -6(x²+4)⁵ .(2x)

dy/dx = 6y⁵ / -6(x² + 4)⁵ .(2x)

dy/dx = 6y⁵ / 12x(x² +4)⁵

Although the answer is ment to have y⁵ as the numerator, not 6y⁵?
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Another Q. [Simplifying result from product rule?]

Answer of the question:

6(x+4)⁵ .(x+2)⁷ + 7(x+2)⁶ . (x+4)⁶

It is possible to simplify the answer to

(x+4)⁶ (x+4)⁵ (13x+40)

Although, i don't see what to do to get this result.


Thanks in advance,
recoil33

 

[lanedance]

thats not quite a question, though reasonably obvious you should explain what you are trying to do

 

[lanedance]

so I am guessing you want to differentiate this implicitly?

Question:

y⁶ + 6 (x²+4)⁶ = 9

6y⁵ .dy/dx . 6(x² + 4)⁵ . (2x) = 0

the first step should be
6y⁵ .dy/dx + 6.6(x² + 4)⁵ . (2x) = 0

 

[recoil33]

Ahh that makes sense, the source from which i got the question had a massive gap between the 6 and (x²+4)⁶ so i did not think to realize they were together.

Yes, i know the answers were obvious, although i could not see where i was going wrong.

Thank you.

 

[lanedance]

i didn't mean the answer was obvious, just that i could work out what you were trying to accomplish

you'll generally get more help if you're clear describing what you are trying to do