`d/(dx)(x^(2) + cosx)^(4)` =

Options

A. `4(x^(2) + cos x)(2x - sin x)`
B. `4(x^(2) - cos x)^(3)(2x - sin x)`
C. `4(x^(2) + cos x)^(3)(2x - sin x)`
D. `4(x^(2) + cos x)^(3)(2x + sin x)`

Answer

To differentiate the function \( y = (x^2 + \cos x)^4 \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
Let \( u = x^2 + \cos x \). Then, we can rewrite \( y \) as:
\[ y = u^4 \]

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = 4u^3 \]

### Step 3: Differentiate the inner function
Now, we need to differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + \cos x) = 2x - \sin x \]

### Step 4: Apply the chain rule
Now, we can apply the chain rule to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4u^3 \cdot (2x - \sin x) \]

### Step 5: Substitute back for \( u \)
Substituting back \( u = x^2 + \cos x \):
\[ \frac{dy}{dx} = 4(x^2 + \cos x)^3 \cdot (2x - \sin x) \]

### Final Answer
Thus, the derivative of \( y = (x^2 + \cos x)^4 \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 4(x^2 + \cos x)^3 (2x - \sin x) \]

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