If `y=(ax+b)^(n)`, they `(dy)/(dx)` is equal to :

Options

A. `(n(ax+b)^(n-1))/a`
B. `n(ax+b)^(n-1)`
C. `na(ax+b)^(n-1)`
D. None of these

Answer

To find the derivative of \( y = (ax + b)^n \) with respect to \( x \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
In the expression \( y = (ax + b)^n \), we can identify:
- The outer function \( u^n \) where \( u = ax + b \)
- The inner function \( u = ax + b \)

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( u^n \) with respect to \( u \) is:
\[ \frac{dy}{du} = n u^{n-1} \]

### Step 3: Differentiate the inner function
The derivative of the inner function \( u = ax + b \) with respect to \( x \) is:
\[ \frac{du}{dx} = a \]

### Step 4: Apply the chain rule
According to the chain rule, the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found in Steps 2 and 3:
\[ \frac{dy}{dx} = n (ax + b)^{n-1} \cdot a \]

### Step 5: Write the final answer
Thus, the derivative of \( y = (ax + b)^n \) with respect to \( x \) is:
\[ \frac{dy}{dx} = n a (ax + b)^{n-1} \]

### Summary
The final answer is:
\[ \frac{dy}{dx} = n a (ax + b)^{n-1} \]

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