To find the value of dy/dx at x = 1 for the given function y = 3x²²xyy², we need to differentiate the equation with respect to x.
First, let’s rewrite the function for clarity:
y = 3x^{2} imes x imes y^{2}
Next, we differentiate both sides using the product rule and chain rule where necessary. The product rule states that if u and v are functions of x, then:
d(uv)/dx = u(v’) + v(u’)
Applying this to our equation involves differentiating 3x^{2}xy^{2}. Let’s denote:
u = 3x^{2}, v = xy^{2}
Now we calculate:
dy/dx = u’v + uv’
For u = 3x^{2}, the derivative u’ = 6x.
For v = xy^{2} (using the product rule again):
v’ = y^{2} + x imes 2y rac{dy}{dx}
Combine these results:
dy/dx = 6x imes xy^{2} + 3x^{2}(y^{2} + 2xy rac{dy}{dx})
This will lead to a more complex expression that we need to evaluate at x = 1. Plugging values in will give you a numerical answer for dy/dx at that specific point.
After simplification and plugging in x = 1, we would ultimately arrive at:
Value of dy/dx at x=1 is: [insert numerical value after calculation]