Question

If y = (cosx²)², then \(\frac{dy}{dx}\) is equal to

• A. -4x sin2x²
• B. -x sinx²
• C. -2x sin2x²
• D. -x cos2x²

Answer 1

The Correct Option is C

Given: \( y = (\cos(x^2))^2 \)
We need to find \( \frac{dy}{dx} \).

Step 1: Simplifying the function
\( y = \cos^2(x^2) \) can be written as:
\[ y = (\cos(u))^2 \quad \text{where} \quad u = x^2 \]
Step 2: Applying chain rule
Using the chain rule, we differentiate \( y = (\cos(u))^2 \). First, differentiate the outer function: \[ \frac{d}{du} \left[ (\cos(u))^2 \right] = 2\cos(u) \cdot (-\sin(u)) \] So, we get: \[ \frac{dy}{du} = -2\cos(u)\sin(u) \] Next, differentiate \( u = x^2 \) with respect to \( x \): \[ \frac{du}{dx} = 2x \]
Step 3: Combining the results
Now applying the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = -2\cos(x^2) \sin(x^2) \times 2x \] Simplifying: \[ \frac{dy}{dx} = -4x \cos(x^2) \sin(x^2) \]
Step 4: Final Answer
The expression simplifies to \( -2x \sin(2x^2) \) (since \( 2\sin(A)\cos(A) = \sin(2A) \)).

The correct answer is (C).