`y=(2x+3)^(4)-(7x-1)^(2)+(2)/((3x+1)^(3))+(4)/((4x-3)^(2))`. Find `(dy)/(dx)`

Explanation

To find the derivative \( \frac{dy}{dx} \) of the function

\[ y = (2x + 3)^4 - (7x - 1)^2 + \frac{2}{(3x + 1)^3} + \frac{4}{(4x - 3)^2}, \]

we will use the chain rule and the power rule of differentiation.

### Step-by-Step Solution:

1. **Differentiate the first term**:
\[ y_1 = (2x + 3)^4 \]
Using the chain rule:
\[ \frac{dy_1}{dx} = 4(2x + 3)^3 \cdot \frac{d}{dx}(2x + 3) = 4(2x + 3)^3 \cdot 2 = 8(2x + 3)^3. \]

2. **Differentiate the second term**:
\[ y_2 = -(7x - 1)^2 \]
Again using the chain rule:
\[ \frac{dy_2}{dx} = -2(7x - 1) \cdot \frac{d}{dx}(7x - 1) = -2(7x - 1) \cdot 7 = -14(7x - 1). \]

3. **Differentiate the third term**:
\[ y_3 = \frac{2}{(3x + 1)^3} = 2(3x + 1)^{-3} \]
Using the chain rule:
\[ \frac{dy_3}{dx} = 2 \cdot (-3)(3x + 1)^{-4} \cdot \frac{d}{dx}(3x + 1) = -6(3x + 1)^{-4} \cdot 3 = -18(3x + 1)^{-4}. \]

4. **Differentiate the fourth term**:
\[ y_4 = \frac{4}{(4x - 3)^2} = 4(4x - 3)^{-2} \]
Again using the chain rule:
\[ \frac{dy_4}{dx} = 4 \cdot (-2)(4x - 3)^{-3} \cdot \frac{d}{dx}(4x - 3) = -8(4x - 3)^{-3} \cdot 4 = -32(4x - 3)^{-3}. \]

5. **Combine all derivatives**:
Now, we can combine all the derivatives:
\[ \frac{dy}{dx} = 8(2x + 3)^3 - 14(7x - 1) - 18(3x + 1)^{-4} - 32(4x - 3)^{-3}. \]

### Final Answer:
Thus, the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = 8(2x + 3)^3 - 14(7x - 1) - 18(3x + 1)^{-4} - 32(4x - 3)^{-3}. \]