**Finding the Right Hand Derivative of f(x-1)=2x^2-3x 1 at k=1**

To find the right-hand derivative of a function, we must take the limit of the difference quotient as x approaches a point from the right-hand side. In this case, we are asked to find the right-hand derivative of f(x-1)=2x^2-3x 1 at k=1. Let's break down the steps to solve this problem.

**Step 1: Rewrite the Function**

To find the right-hand derivative of f(x-1)=2x^2-3x 1 at k=1, we need to rewrite the function with x-k.

f(x-1)=2x^2-3x+1 becomes f(x)=2(x+1)^2-3(x+1)+1

**Step 2: Find the Difference Quotient**

The difference quotient for the right-hand derivative is:

f'(1+) = lim [f(x) - f(1)] / [x - 1]

where the limit is taken as x approaches 1 from the right-hand side.

**Step 3: Evaluate the Difference Quotient**

We can evaluate the difference quotient by plugging in x = 1 + h, where h is a small positive number.

f'(1+) = lim [2(1+h+1)^2 - 3(1+h+1) + 1 - (2(1+1)^2 - 3(1+1) + 1)] / [h]

f'(1+) = lim [2(h+2)^2 - 3(h+2)] / [h]

**Step 4: Simplify the Expression**

We can simplify the expression by expanding the square and simplifying the terms.

f'(1+) = lim [2h^2 + 8h + 8 - 3h - 6] / [h]

f'(1+) = lim [2h^2 + 5h + 2] / [h]

**Step 5: Take the Limit**

To find the right-hand derivative, we need to take the limit of the difference quotient as h approaches 0 from the positive side.

f'(1+) = lim [2h^2 + 5h + 2] / [h] as h approaches 0+

Using L'Hopital's rule, we can take the derivative of the numerator and denominator with respect to h.

f'(1+) = lim [4h + 5] / [1]

f'(1+) = 9

**Step 6: Interpret the Result**

The right-hand derivative of f(x-1)=2x^2-3x 1 at k=1 is 9. This means that the slope of the tangent line at x = 1 from the right-hand side is 9. In other words, as x approaches 1 from the right, the function is increasing at a rate of 9 units per unit change in x.

In conclusion, we have found the right-hand derivative of f(x-1)=2x^2-3x 1 at k=1 by finding the difference quotient and taking the limit as x approaches 1 from the right-hand side. The answer is 9, which represents the slope of the tangent line