`(d)/(dx)((x^(2))/(x-1))=`

Options

A. `1+(1)/((x-1)^(2))`
B. `(1)/((x-1)^(2))-1`
C. `1-(1)/((x-1)^(2))`
D. `(-1)/((x-1)^(2))`

Answer

To solve the problem \(\frac{d}{dx}\left(\frac{x^2}{x-1}\right)\), we will use the quotient rule for differentiation. The quotient rule states that if you have a function \(\frac{u}{v}\), then its derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \(u = x^2\) and \(v = x - 1\).

### Step 1: Identify \(u\) and \(v\)
Let:
- \(u = x^2\)
- \(v = x - 1\)

### Step 2: Find \(\frac{du}{dx}\) and \(\frac{dv}{dx}\)
Now, we need to find the derivatives of \(u\) and \(v\):
- \(\frac{du}{dx} = \frac{d}{dx}(x^2) = 2x\)
- \(\frac{dv}{dx} = \frac{d}{dx}(x - 1) = 1\)

### Step 3: Apply the Quotient Rule
Now, we can apply the quotient rule:
\[ \frac{d}{dx}\left(\frac{x^2}{x-1}\right) = \frac{(x - 1)(2x) - (x^2)(1)}{(x - 1)^2} \]

### Step 4: Simplify the Numerator
Now, let's simplify the numerator:
\[ = \frac{(2x(x - 1)) - x^2}{(x - 1)^2} \]
\[ = \frac{(2x^2 - 2x) - x^2}{(x - 1)^2} \]
\[ = \frac{2x^2 - 2x - x^2}{(x - 1)^2} \]
\[ = \frac{x^2 - 2x}{(x - 1)^2} \]

### Step 5: Factor the Numerator
Now, we can factor the numerator:
\[ = \frac{x(x - 2)}{(x - 1)^2} \]

### Final Answer
Thus, the derivative \(\frac{d}{dx}\left(\frac{x^2}{x-1}\right)\) is:
\[ \frac{x(x - 2)}{(x - 1)^2} \]

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