Differentiate the following w.r.t.x :
`sin^(4)(ax+b)^(2)`

Answer

To differentiate the function \( y = \sin^4(ax + b)^2 \) with respect to \( x \), we will use the chain rule and the product rule. Let's break it down step by step.

### Step 1: Identify the outer and inner functions
We can rewrite the function as:
\[ y = \left( \sin(ax + b) \right)^4 \]
This means we have an outer function \( u^4 \) where \( u = \sin(ax + b) \).

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

### Step 3: Differentiate the inner function
Next, we need to differentiate \( u = \sin(ax + b) \) with respect to \( x \):
\[ \frac{du}{dx} = \cos(ax + b) \cdot \frac{d}{dx}(ax + b) \]
Since \( \frac{d}{dx}(ax + b) = a \), we have:
\[ \frac{du}{dx} = a \cos(ax + b) \]

### Step 4: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 4 \sin^3(ax + b) \cdot (a \cos(ax + b)) \]

### Step 5: Simplify the expression
Putting it all together, we get:
\[ \frac{dy}{dx} = 4a \sin^3(ax + b) \cos(ax + b) \]

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