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

Answer

To find the derivative of the function \( f(x) = \sin^3(ax + b) \), we will apply the chain rule and the power rule. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We can express the function in a clearer form:
\[ f(x) = (\sin(ax + b))^3 \]

### Step 2: Apply the power rule
Using the power rule, the derivative of \( x^n \) is \( n \cdot x^{n-1} \). Here, we will differentiate \( (\sin(ax + b))^3 \):
\[ f'(x) = 3(\sin(ax + b))^2 \cdot \frac{d}{dx}(\sin(ax + b)) \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( \sin(ax + b) \). Using the chain rule:
\[ \frac{d}{dx}(\sin(ax + b)) = \cos(ax + b) \cdot \frac{d}{dx}(ax + b) \]
The derivative of \( ax + b \) is \( a \) (since \( b \) is a constant).

### Step 4: Combine the derivatives
Now we can substitute back into our expression for \( f'(x) \):
\[ f'(x) = 3(\sin(ax + b))^2 \cdot \cos(ax + b) \cdot a \]

### Step 5: Final expression
Thus, the final expression for the derivative is:
\[ f'(x) = 3a \sin^2(ax + b) \cos(ax + b) \]