Answer:

We can find the derivatives of the given functions using the power rule and product rule of differentiation:

1. y = x² + 2x + 1

dy/dx = d/dx (x²) + d/dx (2x) + d/dx (1)

dy/dx = 2x + 2

Therefore, the derivative of y = x² + 2x + 1 is dy/dx = 2x + 2.

2. y = 873

Since y is a constant, its derivative is zero.

Therefore, the derivative of y = 873 is dy/dx = 0.

3. y = (3x⁴+5)^5

Using the chain rule, we can write:

dy/dx = d/dx (3x⁴+5)^5

dy/dx = 5(3x⁴+5)^4 * d/dx (3x⁴+5)

dy/dx = 5(3x⁴+5)^4 * 12x³

Therefore, the derivative of y = (3x⁴+5)^5 is dy/dx = 60x³(3x⁴+5)^4.

4. y = (x+2)(4x+1)

Using the product rule, we can write:

dy/dx = d/dx (x+2) * (4x+1) + (x+2) * d/dx (4x+1)

dy/dx = 1*(4x+1) + (x+2)*4

dy/dx = 4x + 1 + 4x + 8

Therefore, the derivative of y = (x+2)(4x+1) is dy/dx = 8x + 9.

5. y = 5x² + 7x² + 7x³

Simplifying the expression inside the parentheses, we get:

y = 12x² + 7x³

Using the power rule and the constant multiple rule, we can write:

dy/dx = d/dx (12x²) + d/dx (7x³)

dy/dx = 24x + 21x²

Therefore, the derivative of y = 5x² + 7x² + 7x³ is dy/dx = 24x + 21x².