Use the formula for slope of a line tangent to a parametric curve to find dy/dx for the curve c(s) = (s^-1-6s, -7s^3) at the point with s=-4

x = s^-1 - 6s

dx/ds = -s^-2 - 6

y = -7s^-3
dy/ds = 21s^-4

dy/dx = (dy/ds) / (dx/ds)
= (-s^-2 -6)/(21s^-4)
= (-s^-2 -6)/(21s^-4) * (s^4)/s^4)
= ( -s^2 - 6s^4)/21

when s = -4
dy/dx = (-16 - 1536)/21 = -1552/21

Oops. y = -7s^3

dy/ds = -21s^2

Make that fix, then follow the rest of the steps.

Those silly copy errors will do it every time

To find the slope of a line tangent to a parametric curve, we can use the formula:

dy/dx = (dy/ds) / (dx/ds)

where dy/ds and dx/ds represent the derivatives of the y-coordinate and x-coordinate with respect to the parameter s.

Given the parametric curve c(s) = (s^-1 - 6s, -7s^3), and we want to find dy/dx for the point where s = -4.

First, let's find dy/ds and dx/ds:

dy/ds = d(-7s^3) / ds = -21s^2
dx/ds = d(s^-1 - 6s) / ds = d(s^-1)/ds - d(6s)/ds = -s^-2 - 6

Substitute s = -4 into the derivatives to find the values at s = -4:

dy/ds = -21(-4)^2 = -21(16) = -336
dx/ds = -(-4)^-2 - 6 = -(1/16) - 6 = -1/16 - 96/16 = -97/16

Now, use the formula to calculate dy/dx:

dy/dx = (dy/ds) / (dx/ds) = (-336) / (-97/16) = (-336) * (16 / -97) = 5376 / 97

Therefore, dy/dx for the curve c(s) at the point where s = -4 is 5376 / 97.