STATEMENT -1 : for the function y= f(x), `f(x) ,({1+((dy)/dx)^(2)}^(3/2))/((d^(2)y)/(dx^(2))) = - ({1+ (dx/dy)^(2)}^(3/2))/((d^(2)x)/(dy^(2)))`
STATEMENT -2 : `(dy)/(dx) = (1/(dx))/dy and (d^(2)y)/(dx^(2)) = d/dx (dy/(dx))`

Answer

To solve the given problem, we need to analyze both statements and determine their validity step by step.

### Step 1: Analyze Statement 1
We start with the equation given in Statement 1:

\[ \frac{(1 + \left(\frac{dy}{dx}\right)^2)^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = -\frac{(1 + \left(\frac{dx}{dy}\right)^2)^{\frac{3}{2}}}{\frac{d^2x}{dy^2}} \]

#### Step 1.1: Rewrite \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\)
Using the relationships:
\[ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \quad \text{and} \quad \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]

We can rewrite the left-hand side (LHS) of the equation.

#### Step 1.2: Substitute into LHS
Substituting the expressions into LHS, we get:

\[ \text{LHS} = \frac{\left(1 + \left(\frac{1}{\frac{dx}{dy}}\right)^2\right)^{\frac{3}{2}}}{\frac{d}{dx}\left(\frac{dy}{dx}\right)} \]

This simplifies to:

\[ \text{LHS} = \frac{\left(1 + \frac{1}{\left(\frac{dx}{dy}\right)^2}\right)^{\frac{3}{2}}}{\frac{d}{dx}\left(\frac{dy}{dx}\right)} \]

#### Step 1.3: Analyze the Right-Hand Side (RHS)
Now, we analyze the right-hand side (RHS):

\[ \text{RHS} = -\frac{\left(1 + \left(\frac{dx}{dy}\right)^2\right)^{\frac{3}{2}}}{\frac{d^2x}{dy^2}} \]

#### Step 1.4: Compare LHS and RHS
We need to check if LHS equals RHS. However, the negative sign in front of RHS indicates that LHS cannot equal RHS unless both sides are zero, which is generally not the case. Thus, we conclude that Statement 1 is **false**.

### Step 2: Analyze Statement 2
Statement 2 states:

\[ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \quad \text{and} \quad \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]

#### Step 2.1: Verify the First Part
The first part is correct. The derivative \(\frac{dy}{dx}\) is indeed the reciprocal of \(\frac{dx}{dy}\).

#### Step 2.2: Verify the Second Part
The second part is also correct. The second derivative \(\frac{d^2y}{dx^2}\) is defined as the derivative of \(\frac{dy}{dx}\) with respect to \(x\).

### Conclusion
Since both statements are found to be false, we conclude:

- **Statement 1**: False
- **Statement 2**: False