Select the correct alternatives out of given four alternatives in each.
If `y=sin^(4)x+cos^(4)x`, then `(dy)/(dx)=`

Options

A. `4sin^(4)x+4cos^(3)x`
B. `4sin^(3)x-4cos^(3)x`
C. `-sin4x`
D. 0

Answer

To solve the problem, we need to differentiate the function \( y = \sin^4 x + \cos^4 x \) with respect to \( x \). Let's go through the steps systematically.

### Step 1: Differentiate the function
We start with the function:
\[ y = \sin^4 x + \cos^4 x \]
We will differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\sin^4 x) + \frac{d}{dx}(\cos^4 x) \]

### Step 2: Apply the chain rule
Using the chain rule, we differentiate each term:
1. For \( \sin^4 x \):
\[ \frac{d}{dx}(\sin^4 x) = 4 \sin^3 x \cdot \frac{d}{dx}(\sin x) = 4 \sin^3 x \cdot \cos x \]
2. For \( \cos^4 x \):
\[ \frac{d}{dx}(\cos^4 x) = 4 \cos^3 x \cdot \frac{d}{dx}(\cos x) = 4 \cos^3 x \cdot (-\sin x) = -4 \cos^3 x \sin x \]

### Step 3: Combine the derivatives
Now we combine the results from the two derivatives:
\[ \frac{dy}{dx} = 4 \sin^3 x \cos x - 4 \cos^3 x \sin x \]
Factoring out the common terms:
\[ \frac{dy}{dx} = 4 \sin x \cos x (\sin^2 x - \cos^2 x) \]

### Step 4: Use trigonometric identities
We can simplify \( \sin^2 x - \cos^2 x \) using the identity:
\[ \sin^2 x - \cos^2 x = -\cos(2x) \]
Thus, we can rewrite the derivative:
\[ \frac{dy}{dx} = 4 \sin x \cos x (-\cos(2x)) \]
Using the double angle identity for sine:
\[ \sin(2x) = 2 \sin x \cos x \]
We can express \( 4 \sin x \cos x \) as:
\[ 4 \sin x \cos x = 2 \sin(2x) \]
So, we have:
\[ \frac{dy}{dx} = -2 \sin(2x) \cos(2x) \]

### Step 5: Final expression
Using the identity for \( \sin(2x) \cos(2x) \):
\[ \sin(2x) \cos(2x) = \frac{1}{2} \sin(4x) \]
Thus, we can write:
\[ \frac{dy}{dx} = -\sin(4x) \]

### Conclusion
The final result is:
\[ \frac{dy}{dx} = -\sin(4x) \]