Find the derivative s) \( y=\cos ^{4} x+\sin ^{4} x \)

It seems that there was an issue with calculating the derivatives of the expressions directly. However, I can still guide you through the process of finding the derivative of the function \( y = \cos^4 x + \sin^4 x \) step by step. ### Step 1: Use the Chain Rule To differentiate \( y = \cos^4 x + \sin^4 x \), we will apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \). ### Step 2: Differentiate Each Term 1. **Differentiate \( \cos^4 x \)**: - Let \( u = \cos x \), then \( \cos^4 x = u^4 \). - The derivative of \( u^4 \) is \( 4u^3 \cdot \frac{du}{dx} \). - Since \( \frac{du}{dx} = -\sin x \), we have: \[ \frac{d}{dx}(\cos^4 x) = 4\cos^3 x \cdot (-\sin x) = -4\cos^3 x \sin x \] 2. **Differentiate \( \sin^4 x \)**: - Let \( v = \sin x \), then \( \sin^4 x = v^4 \). - The derivative of \( v^4 \) is \( 4v^3 \cdot \frac{dv}{dx} \). - Since \( \frac{dv}{dx} = \cos x \), we have: \[ \frac{d}{dx}(\sin^4 x) = 4\sin^3 x \cdot \cos x \] ### Step 3: Combine the Derivatives Now, we can combine the derivatives of both terms: \[ \frac{dy}{dx} = -4\cos^3 x \sin x + 4\sin^3 x \cos x \] ### Step 4: Factor the Expression We can factor out common terms: \[ \frac{dy}{dx} = 4\cos x \sin x (\sin^2 x - \cos^2 x) \] ### Final Result Thus, the derivative of \( y = \cos^4 x + \sin^4 x \) is: \[ \frac{dy}{dx} = 4\cos x \sin x (\sin^2 x - \cos^2 x) \]