If `y=log_(e) e^(sinx^(2))," find "(dy)/(dx)`

Answer

To find the derivative of the function \( y = \log_e(e^{\sin(x^2)}) \), we can follow these steps:

### Step 1: Simplify the expression
The function can be simplified using the property of logarithms that states \( \log_a(a^b) = b \). Here, we have:
\[ y = \log_e(e^{\sin(x^2)}) = \sin(x^2) \]

### Step 2: Differentiate the simplified function
Now that we have \( y = \sin(x^2) \), we can differentiate it with respect to \( x \). We will use the chain rule for differentiation. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), and we will also need to differentiate the inner function \( u = x^2 \):
\[ \frac{dy}{dx} = \cos(x^2) \cdot \frac{d}{dx}(x^2) \]

### Step 3: Differentiate \( x^2 \)
The derivative of \( x^2 \) is:
\[ \frac{d}{dx}(x^2) = 2x \]

### Step 4: Combine the results
Now, substituting back into our derivative:
\[ \frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x \cos(x^2) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x \cos(x^2) \]

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