# How do you find y'' by implicit differentiation for #4x^3 + 3y^3 = 6#?

Please see below.

Differentiate again, using the quotient rule to get

I prefer to remove the common factor before proceeding:

Now, simplify the complex fraction using your chosen technique.

I see that I can reduce the fraction, but before I do there's a step I can do to simplify a lot.

Way back at the start of the problem, we were told that

Now simplify the quotient:

Note 1 Although we could differentiate again immediately, I prefer not to.

It works, but it's kind of a mess.

Note 2 This step is typical of certain kinds of implicit differentiation second derivative problems. If you remember to look for it, it can simplify the result considerably.

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To find ( y'' ) by implicit differentiation for the equation ( 4x^3 + 3y^3 = 6 ), follow these steps:

- Differentiate both sides of the equation with respect to ( x ) using the chain rule.
- Solve for ( \frac{d^2y}{dx^2} ), which represents ( y'' ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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