DERIVATIVE OF 1+2(X-1)-(X-1)^2+(X-1)^3

To find the derivative of the function 1+2(X-1)-(X-1)^2+(X-1)^3, we will use the power rule which states that the derivative of x^n is n*x^(n-1).

Step 1: Identify the terms in the function. The terms are: 1, 2(X-1), -(X-1)^2, and (X-1)^3.

Step 2: Find the derivative of each term.

• The derivative of a constant, like 1, is 0.

• For the term 2(X-1), treat (X-1) as a single variable. The derivative of X with respect to X is 1, so the derivative of this term is 2.

• For the term -(X-1)^2, use the power rule. The derivative is -2*(X-1). But, we also have to apply the chain rule because of the (X-1) term. The derivative of (X-1) with respect to X is 1, so the overall derivative of this term is -2*(X-1)1 = -2(X-1).

• For the term (X-1)^3, again use the power rule. The derivative is 3*(X-1)^2. Applying the chain rule for the (X-1) term, the overall derivative of this term is 3*(X-1)^21 = 3(X-1)^2.

Step 3: Combine the derivatives of each term to get the derivative of the entire function.

So, the derivative of the function 1+2(X-1)-(X-1)^2+(X-1)^3 is 0 + 2 - 2*(X-1) + 3*(X-1)^2.