Answer:

Step-by-step explanation: 1. y = x² + 2x + 1

y' = d/dx(x²) + d/dx(2x) + d/dx(1)

y' = 2x + 2 + 0

y' = 2x + 2

Therefore, the derivative of y = x² + 2x + 1 is y' = 2x + 2

y = 5x³ + 7x² - 7x + 3

y' = d/dx(5x³) + d/dx(7x²) - d/dx(7x) + d/dx(3)

y' = 15x² + 14x - 7 + 0

y' = 15x² + 14x - 7

Therefore, the derivative of y = 5x³ + 7x² - 7x + 3 is y' = 15x² + 14x - 7

3. y = 3x⁵ - 2x^-⁴ + 4x³ + 5^-² + 3x - 5​

y' = d/dx(3x⁵) - d/dx(2x^-⁴) + d/dx(4x³) + d/dx(5^-²) + d/dx(3x) - d/dx(5)

y' = 15x⁴ + 8x^(-5) + 12x² + 0 + 3 - 0

y' = 15x⁴ + 8x^(-5) + 12x² + 3

Therefore, the derivative of y = 3x⁵ - 2x^-⁴ + 4x³ + 5^-² + 3x - 5​ is y' = 15x⁴ + 8x^(-5) + 12x² + 3

Verified answer

Sure! Here are the derivatives of the given functions:

1. y = x² + 2x + 1

The derivative of y with respect to x is:

dy/dx = 2x + 2

2. y = 5x³ + 7x² - 7x + 3

The derivative of y with respect to x is:

dy/dx = 15x² + 14x - 7

3. y = 3x⁵ - 2x^-⁴ + 4x³ + 5^-² + 3x - 5

The derivative of y with respect to x is:

dy/dx = 15x⁴ + 8x^(-5) + 12x² + 0 + 3

Note: The derivative of a constant (in this case, 5^(-2)) is zero, and the derivative of a term with a negative exponent (in this case, 2x^(-4)) requires the use of the chain rule, resulting in the factor of -8x^(-5).