`sin^(3)(ax+b)`

Answer

To find the differential coefficient of the function \( y = \sin^3(ax + b) \) with respect to \( x \), we will use the chain rule and the power rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have the function:
\[ y = \sin^3(ax + b) \]

### Step 2: Apply the chain rule
To differentiate \( y \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let \( u = \sin(ax + b) \). Thus, we can rewrite \( y \) as:
\[ y = u^3 \]

### Step 3: Differentiate using the power rule
Now, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 3u^2 \]

### Step 4: Differentiate the inner function
Next, we need to differentiate \( u = \sin(ax + b) \) with respect to \( x \):
\[ \frac{du}{dx} = \cos(ax + b) \cdot \frac{d}{dx}(ax + b) = \cos(ax + b) \cdot a \]

### Step 5: Combine using the chain rule
Now, we can combine these results using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot a \cdot \cos(ax + b) \]

### Step 6: Substitute back for \( u \)
Substituting back \( u = \sin(ax + b) \):
\[ \frac{dy}{dx} = 3(\sin(ax + b))^2 \cdot a \cdot \cos(ax + b) \]

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

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