Question

The relation y^2(4-x)=x^2 has a slope of when x=3 and y=-3

Answer 1

The slope of the curve y^2(4-x) = x^2 at the point (3, -3) is 2.5, which is found using implicit differentiation.

Step-by-step explanation:

To find the slope of the curve described by the equation y^2(4-x) = x^2 at the point (x, y) = (3, -3), we need to use implicit differentiation. The slope at any point on a curve is given by the derivative dy/dx, which represents how y changes with respect to x. Since the relation is not that of a straight line, the slope will change at different points on the curve.

First, we apply the product rule for differentiation to the left side of the equation and get:

2y(dy/dx)(4-x) + y^2(-1) = 2x.

Then, we plug in the values x = 3 and y = -3:

-6(dy/dx)(1) - 9 = 6.

Solving for dy/dx, we find:

dy/dx = -(6 + 9) / -6.

dy/dx = 15 / 6 = 2.5.

So, when x = 3 and y = -3, the slope of the curve y^2(4-x) = x^2 is 2.5.

The complete question is: The relation y^2(4-x)=x^2 has a slope of when x=3 and y=-3 is:

Answer 2

The slope of the given relation when x = 3 and y = -3 is -2.5.

To find the slope of the given relation when x = 3 and y = -3, we can use the concept of implicit differentiation.

First, let's rewrite the equation as
`y^2(4 - x) = x^2:`


`y^2(4 - x) = x^2`

Next, we need to differentiate both sides of the equation with respect to x:


`d/dx (y^2(4 - x)) = d/dx (x^2)`

To differentiate y^2(4 - x) with respect to x, we can use the product rule:


`d/dx (y^2(4 - x)) = (d/dx y^2)(4 - x) + y^2(d/dx (4 - x))`

The derivative of y^2 with respect to x is:


`d/dx (y^2) = 2y(dy/dx)`

The derivative of (4 - x) with respect to x is:

d/dx (4 - x) = -1

Now, let's substitute these derivatives back into the equation:


`2y(dy/dx)(4 - x) + y^2(-1) = 2x`

Since we are given x = 3 and y = -3, we can substitute these values into the equation:


`2(-3)(dy/dx)(4 - 3) + (-3)^2(-1) = 2(3)`

Simplifying the equation further:

-6(dy/dx)(1) + 9(-1) = 6

-6(dy/dx) - 9 = 6

-6(dy/dx) = 6 + 9

-6(dy/dx) = 15

Finally, we can solve for the slope (dy/dx) by dividing both sides by -6:

(dy/dx) = 15 / -6

(dy/dx) = -2.5