If ` y=(1)/(1+x^(q-p) +x^(r-p) ) +(1)/( 1+x^(r-q) + x^(p-q)) + (1)/( 1+x^(p-r) +x^(q-r))` then ` (dy)/(dx)=`

Options

A. 0
B. 1
C. `(p+q+r) x^(p+q+r)`
D. None

Answer

To solve the given problem, we need to differentiate the function:

\[ y = \frac{1}{1 + x^{q-p} + x^{r-p}} + \frac{1}{1 + x^{r-q} + x^{p-q}} + \frac{1}{1 + x^{p-r} + x^{q-r}} \]

### Step 1: Rewrite the function
We can rewrite the function \(y\) as:

\[ y = \frac{1}{1 + x^{q-p} + x^{r-p}} + \frac{1}{1 + x^{r-q} + x^{p-q}} + \frac{1}{1 + x^{p-r} + x^{q-r}} \]

### Step 2: Differentiate each term
To find \(\frac{dy}{dx}\), we will differentiate each term separately using the quotient rule. The derivative of \(\frac{1}{u}\) is \(-\frac{1}{u^2} \frac{du}{dx}\).

#### For the first term:
Let \(u_1 = 1 + x^{q-p} + x^{r-p}\)

\[ \frac{du_1}{dx} = (q-p)x^{q-p-1} + (r-p)x^{r-p-1} \]

Thus,

\[ \frac{d}{dx}\left(\frac{1}{u_1}\right) = -\frac{1}{(u_1)^2} \frac{du_1}{dx} = -\frac{(q-p)x^{q-p-1} + (r-p)x^{r-p-1}}{(1 + x^{q-p} + x^{r-p})^2} \]

#### For the second term:
Let \(u_2 = 1 + x^{r-q} + x^{p-q}\)

\[ \frac{du_2}{dx} = (r-q)x^{r-q-1} + (p-q)x^{p-q-1} \]

Thus,

\[ \frac{d}{dx}\left(\frac{1}{u_2}\right) = -\frac{1}{(u_2)^2} \frac{du_2}{dx} = -\frac{(r-q)x^{r-q-1} + (p-q)x^{p-q-1}}{(1 + x^{r-q} + x^{p-q})^2} \]

#### For the third term:
Let \(u_3 = 1 + x^{p-r} + x^{q-r}\)

\[ \frac{du_3}{dx} = (p-r)x^{p-r-1} + (q-r)x^{q-r-1} \]

Thus,

\[ \frac{d}{dx}\left(\frac{1}{u_3}\right) = -\frac{1}{(u_3)^2} \frac{du_3}{dx} = -\frac{(p-r)x^{p-r-1} + (q-r)x^{q-r-1}}{(1 + x^{p-r} + x^{q-r})^2} \]

### Step 3: Combine the derivatives
Now, we can combine the derivatives of all three terms:

\[ \frac{dy}{dx} = -\frac{(q-p)x^{q-p-1} + (r-p)x^{r-p-1}}{(1 + x^{q-p} + x^{r-p})^2} - \frac{(r-q)x^{r-q-1} + (p-q)x^{p-q-1}}{(1 + x^{r-q} + x^{p-q})^2} - \frac{(p-r)x^{p-r-1} + (q-r)x^{q-r-1}}{(1 + x^{p-r} + x^{q-r})^2} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is given by:

\[ \frac{dy}{dx} = -\left[\frac{(q-p)x^{q-p-1} + (r-p)x^{r-p-1}}{(1 + x^{q-p} + x^{r-p})^2} + \frac{(r-q)x^{r-q-1} + (p-q)x^{p-q-1}}{(1 + x^{r-q} + x^{p-q})^2} + \frac{(p-r)x^{p-r-1} + (q-r)x^{q-r-1}}{(1 + x^{p-r} + x^{q-r})^2}\right] \]