To find the slope of the tangent line at a given point on a curve, we need to find the derivative of the curve with respect to x and evaluate it at the given point.
First, rearrange the equation to isolate y:
xy^3 - yx^3 = 6
y(x - x^3) = 6
y = 6 / (x - x^3)
Now, take the derivative of y with respect to x:
dy/dx = d(6 / (x - x^3))/dx
Using the quotient rule,
dy/dx = (0 - 6 * (1 - 3x^2)) / (x - x^3)^2
= -6(1 - 3x^2) / (x - x^3)^2
To find the slope at the point (1,1), substitute x = 1 into the expression for the derivative:
dy/dx = -6(1 - 3(1)^2) / (1 - 1^3)^2
= -6(1 - 3)/ (1 - 1)^2
= -6(-2)/0
The expression is undefined when the denominator is zero, indicating that the slope at the point (1,1) is undefined. This means that the tangent line at that point is vertical.
Find the slope of the tangent line for the curve xy^3-yx^3=6 at the point(1,1)
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