What is the result of implicit differentiation of the equation 8x^4 + 3y^4 = 32?

To find the derivative using implicit differentiation from the equation (8x^4 + 3y^4 = 32), we start by differentiating both sides with respect to (x).

Differentiating the left side:

1. The term (8x^4) differentiates to (32x^3).

2. The term (3y^4) requires the chain rule, which gives us (12y^3 \frac{dy}{dx}).

So, the derivative of the left side is:

[

32x^3 + 12y^3 \frac{dy}{dx}

]

The right side, which is a constant (32), differentiates to 0.

Setting these equal, we get:

[

32x^3 + 12y^3 \frac{dy}{dx} = 0

]

Next, we solve for (\frac{dy}{dx}):

[

12y^3 \frac{dy}{dx} = -32x^3

]

Then, dividing both sides by (12y^3) results in:

[

\frac{dy}{dx} = -\frac{32x