The problem asks us to find the derivative, \(\rm \frac{dy}{dx}\), of the given function \(\rm y=x^{\sec^2x}\times\frac{1}{x^{\tan^2x}}\). To solve this, we first need to simplify the function \(\rm y\) using exponent rules and trigonometric identities before performing differentiation.

Function Simplification: Initial Steps

Let's start by rewriting the given function:

\[ \rm y = x^{\sec^2x} \times \frac{1}{x^{\tan^2x}} \]

We can simplify the term \(\rm \frac{1}{x^{\tan^2x}}\) using the exponent rule that states \(\rm \frac{1}{a^n} = a^{-n}\). Applying this rule:

\[ \rm \frac{1}{x^{\tan^2x}} = x^{-\tan^2x} \]

Now, substitute this simplified form back into the original expression for \(\rm y\):

\[ \rm y = x^{\sec^2x} \times x^{-\tan^2x} \]

Next, we use another fundamental exponent rule for multiplying terms with the same base: \(\rm a^m \times a^n = a^{m+n}\). Applying this rule to our expression:

\[ \rm y = x^{\sec^2x + (-\tan^2x)} \] \[ \rm y = x^{\sec^2x - \tan^2x} \]

At this point, we recall a key trigonometric identity that simplifies the exponent:

• Trigonometric Identity: \(\sec^2x - \tan^2x = 1\)

Substituting this identity into our simplified expression for \(\rm y\):

\[ \rm y = x^1 \] \[ \rm y = x \]

Thus, the seemingly complex function \(\rm y=x^{\sec^2x}\times\frac{1}{x^{\tan^2x}}\) simplifies significantly to just \(\rm y=x\).

Derivative Calculation: Applying Differentiation Rules

Now that we have simplified \(\rm y\) to \(\rm y=x\), finding its derivative \(\rm \frac{dy}{dx}\) is straightforward. We apply the basic power rule of differentiation, which states that for a function \(\rm y=x^n\), its derivative \(\rm \frac{dy}{dx} = nx^{n-1}\). In our simplified function \(\rm y=x\), the value of \(\rm n\) is 1.

\[ \rm \frac{dy}{dx} = \frac{d}{dx}(x) \] Applying the power rule with \(\rm n=1\): \[ \rm \frac{dy}{dx} = 1 \cdot x^{1-1} \] \[ \rm \frac{dy}{dx} = 1 \cdot x^0 \] Since any non-zero number raised to the power of 0 is 1 (\(\rm x^0=1\) for \(\rm x \ne 0\)): \[ \rm \frac{dy}{dx} = 1 \cdot 1 \] \[ \rm \frac{dy}{dx} = 1 \]

Result Analysis: Understanding the Final Derivative

The derivative \(\rm \frac{dy}{dx}\) represents the instantaneous rate of change of \(\rm y\) with respect to \(\rm x\). Since our original function simplifies to \(\rm y=x\), its rate of change with respect to \(\rm x\) is constant and equal to 1. This means that for every unit increase in \(\rm x\), \(\rm y\) also increases by one unit, indicating a direct and linear relationship.

The final answer is \(\rm 1\).