To determine the value of dy/dx when given the expression 3x^2y^2, we need to perform implicit differentiation, assuming there is a relationship between x and y.
First, let’s set the expression equal to a constant, say C 3x^2y^2 = C. Now we will differentiate both sides of this equation with respect to x.
Using the product rule and the chain rule, we differentiate:
1. Differentiate 3x^2 with respect to x:
6xy^2
2. Differentiate y^2 with respect to x:
Here, we apply the chain rule: 2y(dy/dx)
Combining these results, applying the product rule, we get:
3(2xy^2)(dx/dx) + 3x^2(2y(dy/dx)) = 0
Thus, our equation becomes:
6xy^2 + 6x^2y(dy/dx) = 0
Now, we can solve for dy/dx:
6x^2y(dy/dx) = -6xy^2
Dividing both sides by 6x^2y (assuming y ≠ 0 and x ≠ 0), we find:
dy/dx = -xy/y^2
Finally, the value of dy/dx depends on the specific values of x and y. If x = x1, substitute this value back into our equation to find the corresponding dy/dx at that point. Simply plug in your known values of x1 and y to obtain the numerical value of the derivative.