Find the derivative of `y= sin x^ 4`.

Answer

To find the derivative of the function \( y = \sin(x^4) \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
The given function is:
\[ y = \sin(x^4) \]

### Step 2: Use substitution
Let us set:
\[ u = x^4 \]
Then, we can rewrite the function in terms of \( u \):
\[ y = \sin(u) \]

### Step 3: Differentiate \( u \) with respect to \( x \)
Now, we need to find the derivative of \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^4) = 4x^3 \]

### Step 4: Differentiate \( y \) with respect to \( u \)
Next, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{d}{du}(\sin(u)) = \cos(u) \]

### Step 5: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(u) \cdot 4x^3 \]

### Step 6: Substitute back for \( u \)
Now, we substitute back \( u = x^4 \):
\[ \frac{dy}{dx} = \cos(x^4) \cdot 4x^3 \]

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

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