If `y=f(x^(2)) and f'(x) =sin x^(2)." Find "(dy)/(dx)`

Answer

To solve the problem where \( y = f(x^2) \) and \( f'(x) = \sin(x^2) \), we need to find \( \frac{dy}{dx} \). Here’s the step-by-step solution:

### Step 1: Differentiate \( y \) with respect to \( x \)
Given \( y = f(x^2) \), we will use the chain rule to differentiate \( y \).

Using the chain rule:
\[ \frac{dy}{dx} = f'(x^2) \cdot \frac{d}{dx}(x^2) \]

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

### Step 3: Substitute \( f'(x^2) \)
We know from the problem statement that \( f'(x) = \sin(x^2) \). Therefore, we substitute \( x^2 \) into \( f' \):
\[ f'(x^2) = \sin((x^2)^2) = \sin(x^4) \]

### Step 4: Combine the results
Now we can combine the results from Step 1, Step 2, and Step 3:
\[ \frac{dy}{dx} = f'(x^2) \cdot 2x = \sin(x^4) \cdot 2x \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x \sin(x^4) \]

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