I saw this question asked, but I didn't understand the answer format. My prof doesn't use dx and dy after he has differentated the problem. He just uses (for example) 8x+2+y+xy(prime)
The question is this:
4x^2+2x+xy=12 and y(2)=-4 find y(prime)(2) by implicit differentiation.
To find y'(2) using implicit differentiation, we need to differentiate both sides of the equation 4x^2 + 2x + xy = 12 with respect to x.
Let's go step by step:
1. Start with the given equation: 4x^2 + 2x + xy = 12.
2. Differentiate both sides of the equation with respect to x. Remember that when differentiating y, we treat it as a function of x. The derivative of a constant (like 12) is zero. The product rule is used to differentiate xy.
d/dx (4x^2) + d/dx (2x) + d/dx (xy) = d/dx (12).
Simplify each term:
8x + 2 + y + x(dy/dx) + y'x = 0 (since the derivative of a constant is zero).
3. Rearrange the equation by combining like terms:
x(dy/dx) + y'x + 8x + y = -2.
4. Now, we want to find y'(2) at x = 2 and y = -4. Substitute these values into the equation obtained in step 3:
(2)(dy/dx) + y'(2)(2) + 8(2) + (-4) = -2.
Simplify and solve for y'(2):
2(dy/dx) + 2y'(2) + 16 - 4 = -2.
2(dy/dx) + 2y'(2) + 12 = -2.
2(dy/dx) + 2y'(2) = -14.
dy/dx + y'(2) = -7.
y'(2) = -7 - dy/dx.
Therefore, to find y'(2), you need to also know the value of dy/dx in order to substitute it into the equation.