[Implicit Differentiation]

(6+y)^3 + 4y = x + 55 with y' at (1, -2)

y' at (1,-2) =?

I got this far but I'm not sure if I'm doing it right: 3(6+y)^2(1dy/dx)+4dy/dx=1

good so far, now sub in the point

3(6-2)^2(dy/dx) + 4dy/dx = 1
148dy/dx = 1
dy/dx = 1/144

Not correct.

then surely you can find my arithmetic error, I just noticed it too.

To find y' (the derivative of y with respect to x) at the point (1, -2) for the equation (6+y)^3 + 4y = x + 55, you're on the right track using implicit differentiation. Here's the step-by-step process:

Step 1: Differentiate both sides of the equation with respect to x.
The derivative of (6+y)^3 + 4y with respect to x can be written as follows:
3(6+y)^2 * (1+dy/dx) + 4(dy/dx) = 1

Now, you correctly applied the chain rule by multiplying the derivative of the outer function (6+y)^3 by the derivative of the inner function (dy/dx), denoted as (1+dy/dx). However, I can see that you may have missed distributing the derivative when differentiating the term 4y.

To differentiate 4y with respect to x, you will need to apply the product rule since it involves the product of two functions: 4 and y. The derivative of the product of two functions u(x)v(x) with respect to x is given by u(x)v'(x) + v(x)u'(x).

So, applying the product rule to 4y, the derivative becomes 4(dy/dx) + y * (d(4)/dx).
Since d(4)/dx = 0 (since 4 is a constant), the term y * (d(4)/dx) becomes 0, simplifying the equation to:
3(6+y)^2 * (1+dy/dx) + 4(dy/dx) = 1

Step 2: Substitute the point (1, -2) into the equation.
Substitute x = 1 and y = -2 into the equation obtained from Step 1:
3(6+(-2))^2 * (1+dy/dx) + 4(dy/dx) = 1
3(4)^2 * (1+dy/dx) + 4(dy/dx) = 1
3(16) * (1+dy/dx) + 4(dy/dx) = 1
48(1+dy/dx) + 4(dy/dx) = 1

Step 3: Solve the equation for dy/dx.
Combine like terms:
48 + 48(dy/dx) + 4(dy/dx) = 1 + 0
52(dy/dx) = 1 - 48
52(dy/dx) = -47
dy/dx = -47/52
dy/dx = -23/26

Therefore, y' (the derivative of y with respect to x) at the point (1, -2) is -23/26.