how to find the slope of the curve 17x^2+12xy=100-8y^2 at the point where y=1.
-i already d/dx but i didn't know how to find the slope of curve bcus they only give y.
d/dx=(-34x-12y)/(12x+16y)
Given y=1, substitute in original equation and solve for x:
17x^2+12xy=100-8y^2, or
17*x^2+12*x-92=0
which gives
x=-46/17 or x=2
So substitute
(-46/17,1) and (2,1) into your expression for dy/dx to get the numerical value.
To find the slope of the curve at a specific point, you need to find the derivative of the equation with respect to x, substitute the given value for y, and then evaluate the derivative at that point.
Here's how you can find the slope of the curve 17x^2 + 12xy = 100 - 8y^2 at the point where y = 1:
1. Take the derivative of the equation with respect to x. Since you have already found the derivative as (-34x - 12y)/(12x + 16y), let's use that. However, it seems there might be a mistake in the expression you provided. The correct derivative should be:
dy/dx = (-34x - 12y)/(24x + 16y)
2. Next, substitute y = 1 into the equation. This will give you the derivative at the point where y = 1.
dy/dx = (-34x - 12(1))/(24x + 16(1))
dy/dx = (-34x - 12)/(24x + 16)
Now you have the derivative dy/dx in terms of x. To find the slope of the curve at the point where y = 1, evaluate this derivative expression by plugging in the value of x.
If you have a specific value for x, substitute it into the equation dy/dx = (-34x - 12)/(24x + 16). For example, if x = 2:
dy/dx = (-34(2) - 12)/(24(2) + 16)
dy/dx = (-68 - 12)/(48 + 16)
dy/dx = -80/64
dy/dx = -5/4
Therefore, the slope of the curve at the point where y = 1 and x = 2 is -5/4. You can substitute any other value of x to find the slope at a different point on the curve.