Differentiate `4x^4 + 4 cos x` w.r.t `x`

Answer

To differentiate the function \( y = 4x^4 + 4 \cos x \) with respect to \( x \), we will use the rules of differentiation.

### Step-by-Step Solution:

1. **Identify the Function**:
We have the function:
\[ y = 4x^4 + 4 \cos x \]

2. **Apply the Sum Rule**:
The derivative of a sum of functions is the sum of their derivatives. Therefore, we can differentiate each term separately:
\[ \frac{dy}{dx} = \frac{d}{dx}(4x^4) + \frac{d}{dx}(4 \cos x) \]

3. **Differentiate the First Term**:
For the term \( 4x^4 \), we use the power rule:
\[ \frac{d}{dx}(x^n) = n x^{n-1} \]
Here, \( n = 4 \):
\[ \frac{d}{dx}(4x^4) = 4 \cdot 4x^{4-1} = 16x^3 \]

4. **Differentiate the Second Term**:
For the term \( 4 \cos x \), we use the derivative of cosine:
\[ \frac{d}{dx}(\cos x) = -\sin x \]
Thus:
\[ \frac{d}{dx}(4 \cos x) = 4 \cdot (-\sin x) = -4 \sin x \]

5. **Combine the Results**:
Now, we can combine the derivatives of both terms:
\[ \frac{dy}{dx} = 16x^3 - 4 \sin x \]

### Final Answer:
Thus, the derivative of \( 4x^4 + 4 \cos x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 16x^3 - 4 \sin x \]