find the point on the line y=2x-1 which is closest to the point (2,-1).

I used distAnce formula and got
D= (square root 5x^2 -4x +4)
I did....
D=square root (2-x)^2 + (-1-(2x-1))^2

What do I do now?

The point would be the intersection of y = 2x-1 with the perpendicular to that line passing through (2,-1)

slope of y = 2x-1 is 2
so the slope of the perpendicuar is -1/2
and the perpendicular is
y = (-1/2)x + b, with (2,-1) on it
so -1 = (-1/2)(2) + b
b =0

so intersect y = 2x-1 with y = (-1/2)x
2x - 1 = (-1/2)x
4x - 2 = -x
5x = 2
x = 2/5
then y = (-1/2)(2/5) = -1/5

the point is (2/5 , -1/5)

Now that you have the formula for the distance (D), you can differentiate it with respect to the variable x in order to find the minimum distance. Taking the derivative will allow you to find the critical points, where the derivative equals zero or is undefined.

To differentiate the equation for D, you can use the chain rule and the power rule. Let's go step by step:

First, you need to apply the power rule to differentiate the square root term. The derivative of √(2-x)^2 is:

d/dx (√(2-x)^2) = (1/2)(2-x)^2)^(-1/2) * d/dx (2-x)^2

Next, you differentiate the square term (2-x)^2 using the chain rule. The derivative of (2-x)^2 is:

d/dx (2-x)^2 = 2(2-x) * d/dx (2-x)

Now, you can substitute this result back into the derivative of the square root term:

d/dx (√(2-x)^2) = (1/2)(2-x)^2)^(-1/2) * 2(2-x) * d/dx (2-x)

Simplifying this expression further, you have:

d/dx (√(2-x)^2) = (2(2-x)) / (2√(2-x)^2)

Simplifying the fraction, you get:

d/dx (√(2-x)^2) = (2(2-x)) / (2√(2-x)^2) = (2-x) / √(2-x)^2

Now, you can simplify the second term of the distance formula, (-1 - (2x-1))^2:

(-1 - (2x-1))^2 = (-1 - 2x + 1)^2 = (-2x)^2 = 4x^2

Now you can rewrite the distance formula, substituting the derivatives and simplifying the expression:

D = √(2-x)^2 + (-1 - (2x-1))^2
D = √(2-x)^2 + 4x^2
D = √(4 - 4x + x^2) + 4x^2
D = √(x^2 - 4x + 4) + 4x^2

At this point, you can find the critical points by setting the derivative of D equal to zero and solving for x:

d/dx (D) = 0

By differentiating D with respect to x and setting it equal to zero, you can find the values of x that minimize the distance D.