Let the derivative of `f(x)` be defined as `D^(**)f(x)=lim_(hrarr0)(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. (a)`(D^(**)u)v+(D^(**)v)u`
B. (b)`u^(2)(D^(**)v)+v^(2)(D^(**)u)`
C. (c)`D^(**)u+D^(**)v`
D. (d)`uvD^(**)(u+v)`

Answer

To find the value of \( D^{**}(u \cdot v) \) where \( u = f(x) \) and \( v = g(x) \), we start with the definition of the derivative given in the problem:

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

where \( f^{(2)}(x) = (f(x))^2 \).

### Step 1: Write the expression for \( D^{**}(u \cdot v) \)

Using the definition, we have:

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

### Step 2: Expand the square in the numerator

Using the identity \( a^2 - b^2 = (a-b)(a+b) \):

Let \( a = f(x+h) \cdot g(x+h) \) and \( b = f(x) \cdot g(x) \):

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

### Step 3: Simplify \( (f(x+h) \cdot g(x+h) - f(x) \cdot g(x)) \)

Using the product rule:

\[ f(x+h) \cdot g(x+h) - f(x) \cdot g(x) = f(x+h) \cdot g(x+h) - f(x) \cdot g(x+h) + f(x) \cdot g(x+h) - f(x) \cdot g(x) \]

This can be rewritten as:

\[ = (f(x+h) - f(x)) \cdot g(x+h) + f(x) \cdot (g(x+h) - g(x)) \]

### Step 4: Substitute back into the limit

Now substituting back into the limit gives:

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

### Step 5: Apply the limit

As \( h \to 0 \):

1. \( f(x+h) \to f(x) \)
2. \( g(x+h) \to g(x) \)

Thus, we can express the limit as:

\[ D^{**}(u \cdot v) = \lim_{h \to 0} \left( \frac{(f(x+h) - f(x))}{h} \cdot g(x) + f(x) \cdot \frac{(g(x+h) - g(x))}{h} \right) \cdot (2f(x)g(x)) \]

This leads to:

\[ D^{**}(u \cdot v) = 2f(x)g(x) \left( D^{*}f(x) + D^{*}g(x) \right) \]

### Conclusion

Thus, we can conclude that:

\[ D^{**}(u \cdot v) = u^2 D^{*}v + v^2 D^{*}u \]

### Final Answer

The value of \( D^{**}(u \cdot v) \) is:

\[ D^{**}(u \cdot v) = u^2 D^{*}v + v^2 D^{*}u \]