Let the derivative of `f(x)` be defined as `D^(**)f(x)=lim_(hto0)(f^(2)x+h-f^(2)(x))/(h),` where `f^(2)(x)={f(x)}^(2)`.
If `u=f(x),v=g(x)`, then the value of `D^(**)((u)/(v))` is.

Options

A. `(u^(2)(D^(**)v)-v^(2)(D^(**)u))/(v^(4))`
B. `(u(D^(**)v)-v(D^(**)u))/(v^(2))`
C. `(v^(2)(D^(**)u)-u^(2)(D^(**)v))/(v^(4))`
D. `(v(D^(**)u)-u(D^(**)v))/(v^(2))`

Answer

To solve the problem, we need to find the value of \( D^{**} \left( \frac{u}{v} \right) \), where \( u = f(x) \) and \( v = g(x) \). The derivative \( D^{**} f(x) \) is defined as:

\[ D^{**} f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h} \]

### Step-by-Step Solution:

1. **Express the derivative of \( \frac{u}{v} \)**:
We start by applying the definition of the derivative to \( \frac{u}{v} \):
\[ D^{**} \left( \frac{u}{v} \right) = \lim_{h \to 0} \frac{\left( \frac{u}{v} \right)^2 (x + h) - \left( \frac{u}{v} \right)^2 (x)}{h} \]

2. **Rewrite the expression**:
Using the property of limits, we can rewrite the expression:
\[ = \lim_{h \to 0} \frac{\frac{u^2(x + h)}{v^2(x + h)} - \frac{u^2(x)}{v^2(x)}}{h} \]

3. **Combine the fractions**:
Combine the fractions in the numerator:
\[ = \lim_{h \to 0} \frac{u^2(x + h) v^2(x) - u^2(x) v^2(x + h)}{h \cdot v^2(x + h) v^2(x)} \]

4. **Factor the numerator**:
We can factor the numerator:
\[ = \lim_{h \to 0} \frac{(u^2(x + h) - u^2(x)) v^2(x) + u^2(x)(v^2(x) - v^2(x + h))}{h \cdot v^2(x + h) v^2(x)} \]

5. **Apply the limit**:
Now we can apply the limit:
\[ = \frac{v^2(x) \cdot D^{**}(u^2) - u^2(x) \cdot D^{**}(v^2)}{v^4(x)} \]

6. **Final expression**:
Therefore, we have:
\[ D^{**} \left( \frac{u}{v} \right) = \frac{v^2(x) D^{**}(u) - u^2(x) D^{**}(v)}{v^4(x)} \]

### Conclusion:
Thus, the value of \( D^{**} \left( \frac{u}{v} \right) \) is given by:
\[ D^{**} \left( \frac{u}{v} \right) = \frac{g^2(x) D^{**}(f) - f^2(x) D^{**}(g)}{g^4(x)} \]