Question

If 4x² 2x xy=2 and y(2)=−9, find y′(2) by implicit differentiation.

Answer 1

To find y'(2), we used implicit differentiation on the given equation and substituted the values x = 2 and y = -9, resulting in y'(2) = 5.

Step-by-step explanation:

The question requires finding the derivative of y with respect to x, or y', at the point where x = 2, given the equation 4x² + 2x + xy = 2 and that y(2) = -9.

To solve this, we first use implicit differentiation. Differentiating both sides of the given equation with respect to x, we get:

8x + 2 + (x*y' + y) = 0

Now, solving for y' gives:

y' = -(8x + 2) / x - y

Substituting x = 2 and y = -9 into our derivative, we find:

y'(2) = -[(8 * 2) + 2] / 2 - (-9)

The calculation yields y'(2) = 5.