Differentiate the following functions:
`(ax)^(m) + b^(m)`

Answer

To differentiate the function \( (ax)^m + b^m \), we will follow the differentiation rules step by step.

### Step-by-Step Solution:

1. **Identify the function to differentiate**:
\[ y = (ax)^m + b^m \]

2. **Differentiate the first term** \( (ax)^m \):
- We can use the chain rule and the power rule for differentiation. The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).
- Here, we treat \( ax \) as a single variable. Thus, we can write:
\[ \frac{d}{dx}((ax)^m) = m \cdot (ax)^{m-1} \cdot \frac{d}{dx}(ax) \]
- Now, differentiate \( ax \):
\[ \frac{d}{dx}(ax) = a \]
- Therefore, substituting back, we have:
\[ \frac{d}{dx}((ax)^m) = m \cdot (ax)^{m-1} \cdot a = ma(ax)^{m-1} \]

3. **Differentiate the second term** \( b^m \):
- Since \( b \) is a constant, the derivative of a constant is 0:
\[ \frac{d}{dx}(b^m) = 0 \]

4. **Combine the results**:
- Now, we can combine the derivatives of both terms:
\[ \frac{dy}{dx} = ma(ax)^{m-1} + 0 \]
- Thus, the final result is:
\[ \frac{dy}{dx} = ma(ax)^{m-1} \]

### Final Answer:
\[ \frac{dy}{dx} = ma(ax)^{m-1} \]