derivative of `cos(x^4)`

Answer

To find the derivative of the function \( f(x) = \cos(x^4) \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative \( f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- The outer function is \( f(u) = \cos(u) \) where \( u = x^4 \).
- The inner function is \( g(x) = x^4 \).

2. **Differentiate the outer function**:
- The derivative of \( f(u) = \cos(u) \) is \( f'(u) = -\sin(u) \).

3. **Differentiate the inner function**:
- The derivative of \( g(x) = x^4 \) is \( g'(x) = 4x^3 \).

4. **Apply the chain rule**:
- According to the chain rule, the derivative of \( f(g(x)) \) is given by:
\[ f'(g(x)) \cdot g'(x) = -\sin(g(x)) \cdot g'(x) \]
- Substituting \( g(x) \) back in:
\[ f'(x) = -\sin(x^4) \cdot (4x^3) \]

5. **Combine the results**:
- Therefore, the derivative of \( \cos(x^4) \) is:
\[ f'(x) = -4x^3 \sin(x^4) \]

### Final Answer:
The derivative of \( \cos(x^4) \) is:
\[ f'(x) = -4x^3 \sin(x^4) \]

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