Finding the Derivative of f(x) = x^2/(x-1) Explained

Let's examine a function that requires the quotient rule for differentiation.Here's our function: f of x equals x squared divided by x minus one.We can break this function into two parts: g of x equals x squared in the numerator, and h of x equals x minus one in the denominator.Since we're dealing with a fraction, we'll need to use the quotient rule for derivatives.Let's analyze the key characteristics of this function.This function has some interesting behavior, particularly near x equals one, where the denominator becomes zero.Now that we understand the structure of our function, we're ready to apply the quotient rule.To find the derivative, we'll identify the parts of our function that we'll use in the quotient rule.First, let's find the derivatives of g of x and h of x.Now we'll substitute these into the quotient rule formula.Let's substitute our values into the formula. The numerator will have x minus one times two x, minus x squared times one.Next, we'll distribute terms in the numerator. This gives us two x squared minus two x minus x squared.Finally, we can combine like terms in the numerator. Two x squared minus x squared is x squared, and we keep the negative two x.Let's verify our simplification. Two x squared minus x squared equals x squared, and negative two x remains unchanged.Now that we have our simplified derivative, we can move on to analyzing its domain and verifying our result.Now that we have our derivative, let's verify our result and check the domain restrictions.The most important consideration is the domain restriction. When x equals 1, we have a vertical asymptote.Here's our original function in blue. Notice how it approaches infinity as x approaches 1 from either side.And here's our derivative in red. It also has a vertical asymptote at x equals 1, which makes sense given the original function's behavior.Let's go through our verification steps to ensure our derivative is correct.Our final answer is the derivative with its domain restriction. This form is both simplified and properly restricted.