Differential coefficient of `(x^((l+m)/(m-n)))^(1//(n-l))*(x^((m+n)/(n-l)))^(1//(l-m))*(x^((n+l)/(l-m)))^(1//(m-n))` wrt `x` is

Options

A. 1
B. 0
C. -1
D. `x^(lmn)`

Answer

To find the differential coefficient of the expression

\[ \left(x^{\frac{l+m}{m-n}}\right)^{\frac{1}{n-l}} \cdot \left(x^{\frac{m+n}{n-l}}\right)^{\frac{1}{l-m}} \cdot \left(x^{\frac{n+l}{l-m}}\right)^{\frac{1}{m-n}} \]

with respect to \(x\), we can follow these steps:

### Step 1: Simplify the Expression
We start by simplifying the expression. The expression can be rewritten as:

\[ x^{\frac{l+m}{m-n} \cdot \frac{1}{n-l}} \cdot x^{\frac{m+n}{n-l} \cdot \frac{1}{l-m}} \cdot x^{\frac{n+l}{l-m} \cdot \frac{1}{m-n}} \]

Combining the powers of \(x\):

\[ x^{\left(\frac{l+m}{(m-n)(n-l)} + \frac{m+n}{(n-l)(l-m)} + \frac{n+l}{(l-m)(m-n)}\right)} \]

### Step 2: Find a Common Denominator
The common denominator for the three fractions is \((m-n)(n-l)(l-m)\). Therefore, we rewrite each term with this common denominator:

1. The first term becomes:
\[ \frac{(l+m)(l-m)}{(m-n)(n-l)(l-m)} \]

2. The second term becomes:
\[ \frac{(m+n)(m-n)}{(n-l)(l-m)(m-n)} \]

3. The third term becomes:
\[ \frac{(n+l)(n-l)}{(l-m)(m-n)(n-l)} \]

### Step 3: Combine the Numerators
Now we combine the numerators:

\[ (l+m)(l-m) + (m+n)(m-n) + (n+l)(n-l) \]

### Step 4: Expand and Simplify the Numerator
Expanding each term:

1. \((l+m)(l-m) = l^2 - m^2\)
2. \((m+n)(m-n) = m^2 - n^2\)
3. \((n+l)(n-l) = n^2 - l^2\)

Combining these gives:

\[ (l^2 - m^2) + (m^2 - n^2) + (n^2 - l^2) = 0 \]

### Step 5: Final Expression
Since the numerator simplifies to 0, we have:

\[ x^{\frac{0}{(m-n)(n-l)(l-m)}} = x^0 = 1 \]

### Step 6: Differentiate
The differential coefficient of a constant (1) with respect to \(x\) is:

\[ \frac{d}{dx}(1) = 0 \]

### Conclusion
Thus, the differential coefficient of the given expression with respect to \(x\) is:

\[ \boxed{0} \]