If 5x^2+3x+xy=2 andy=-12, find y'(2) by implicit differentiation.

I get

10x + 3 + x(dy/dx) + y = 0
dy/dx = -(10x + 3 +y)/x

so when y = -12 your original equation becomes
5x^2 - 9x - 2 = 0
(5x+1)(x-2) = 0
x = -1/5 or x = 2
so we want dy/dx when x=2 and y = -12

plug those values in and you got it.

To find y'(2) by implicit differentiation, we need to differentiate both sides of the given equation with respect to x. Let's go step by step:

Step 1: Differentiate both sides of the equation with respect to x.
For the left-hand side, we use the product rule for differentiating the term "xy":
(d/dx)(xy) = x(dy/dx) + y(dx/dx) = x(dy/dx) + y

The right-hand side is a constant, so its derivative is zero.

Step 2: Rewrite the equation in terms of y' (dy/dx).
The given equation is 5x^2 + 3x + xy = 2.
Differentiating both sides with respect to x, we have:
d/dx(5x^2 + 3x + xy) = d/dx(2)
10x + 3 + x(dy/dx) + y = 0
x(dy/dx) + 10x + y = -3

Step 3: Solve the equation for dy/dx (y').
Rearranging the terms, we get:
x(dy/dx) = -10x - y - 3

Step 4: Substitute the value of y.
Given that y = -12, we can replace y with -12 in the equation:
x(dy/dx) = -10x - (-12) - 3
x(dy/dx) = -10x + 12 - 3
x(dy/dx) = -10x + 9

Step 5: Solve for dy/dx (y').
To solve for dy/dx, divide both sides of the equation by x:
dy/dx = (-10x + 9) / x

Step 6: Evaluate y'(2).
To find y'(2), we substitute x = 2 into the equation we derived in step 5:
dy/dx = (-10(2) + 9) / (2)
dy/dx = (-20 + 9) / 2
dy/dx = -11 / 2

Therefore, y'(2) = -11/2.