Let me try to ask this question a different way:
4x^2+2x+xy=12 and y(2)=-4 find y(prime)(2) by implicit differentiation.
I can differentiate it to
8x+2+y+xy(prime)=0
I don't know what to do with the y(2)=-4 after this point.
To find y'(2) using implicit differentiation, you can follow these steps:
Step 1: Differentiate both sides of the equation with respect to x. Treat y as a function of x and apply the chain rule where applicable.
Differentiating 4x^2+2x+xy=12 with respect to x, we get:
8x + 2 + y + x(dy/dx) + y'x = 0
Step 2: Simplify the differentiated equation by collecting like terms and isolating the unknown derivative term, dy/dx or y'.
In this case, we need to isolate x(dy/dx) or x(y'), so we can rewrite the equation as:
x(dy/dx) + y'x = -(8x + 2 + y)
Step 3: Substitute the known value y(2) = -4 into the simplified equation.
When x = 2, the equation becomes:
2(dy/dx) + 2y' = -(8(2) + 2 + (-4))
2(dy/dx) + 2y' = -14
Step 4: Solve the equation for dy/dx or y'.
To isolate the derivative term, we divide the equation by 2:
dy/dx + y' = -7
Step 5: Substitute the value x = 2 into the equation.
When x = 2, we get:
dy/dx + y' = -7
Step 6: Solve for y'(2) by substituting the known value y(2) = -4 into the equation from Step 5.
Substituting y(2) = -4 into the equation, we have:
dy/dx + y' = -7
dy/dx + y'(2) = -7
-4 + y'(2) = -7
y'(2) = -7 + 4
y'(2) = -3
Therefore, the value of y'(2) is -3.