Question

Find dx-dy at the point (−8,1) if xy=x+4 / 32y

Answer 1

To find dx-dy at the point (−8,1) in the equation xy=x+4 / 32y, we need to find the partial derivatives with respect to x and y and evaluate them at the given point. The partial derivative with respect to x is -31/8 and the partial derivative with respect to y is -1/4, so dx-dy at the point (-8,1) is -31/8 - (-1/4).

Step-by-step explanation:

To find dx-dy at the point (-8,1) if xy=x+4 / 32y, we need to take partial derivatives with respect to x and y and evaluate them at the given point. Let's start by taking the partial derivative with respect to x:

∂/∂x (xy) = ∂/∂x (x+4 / 32y)

Applying the product rule, we get:

(1)y + x(∂y/∂x) = (1)(32y)(∂/∂x(x+4))

Simplifying, we have: y + x(∂y/∂x) = 32(∂x/∂x)

At the point (-8,1), plug in the values:

1 + (-8)(∂y/∂x) = 32(1)

Now, solve for ∂y/∂x:

-8(∂y/∂x) = 31

∂y/∂x = -31/8

Next, take the partial derivative with respect to y:

∂/∂y (xy) = ∂/∂y (x+4 / 32y)

Applying the product rule, we get:

x + y(∂x/∂y) = (x+4)(∂/∂y(32y))

Simplifying, we have: x + y(∂x/∂y) = (x+4)(32)(∂y/∂y)

At the point (-8,1), plug in the values:

-8 + 1(∂x/∂y) = (x+4)(32)(1)

Now, solve for ∂x/∂y:

∂x/∂y = -8/32

∂x/∂y = -1/4

Therefore, dx-dy at the point (-8,1) is -31/8 - (-1/4).