Application of Derivatives

We are given that a point is moving along the curve described by the equation y=x3−3x2+2x−1 and that the y-coordinate of the point is increasing at a rate of 6 units per second. We need to determine the rate of chang of the x-coordinate of the point when the point is at (2,−1).
Steps to Find the Rate of Change of the x-coordinate:
Differentiating the Curve Equation:
Given y=x3−3x2+2x−1, differentiate y with respect to t to express ‌

dy

dt

:
‌

dy

dt

=‌

d

dt

(x3−3x2+2x−1)=(3x2−6x+2)‌

dx

dt

Using Given Information:
We know ‌

dy

dt

=6. Substitute this value into the differentiated equation:
(3x2−6x+2)‌

dx

dt

=6
Solving for ‌

dx

dt

:
Rearrange the equation to solve for ‌

dx

dt

:
‌

dx

dt

=‌

6

3x2−6x+2

Substitute the Point (2,−1) :
Substitute x=2 into the equation to find ‌

dx

dt

at this specific point:
‌

dx

dt

=‌

6

3(2)2−6(2)+2

=‌

6

3×4−6×2+2

Simplify the Expression:
‌

dx

dt

=‌

6

12−12+2

=‌

6

2

=3
Therefore, the rate of change of the x-coordinate of the point when at (2,−1) is 3 units per second.