Answer:

1. To find the derivative of f(x) = ln(x² + 1), we can apply the chain rule. The chain rule states that if we have a function g(h(x)), the derivative of g(h(x)) with respect to x is given by g'(h(x)) * h'(x).

Let's find the derivative step by step:

f(x) = ln(x² + 1)

f'(x) = (1 / (x² + 1)) * (2x)

= 2x / (x² + 1)

2. To find the derivative of f(x) = (x² + 2x + 1)⁴, we can use the power rule. The power rule states that if we have a function f(x) = (g(x))^n, then the derivative of f(x) is given by n * (g(x))^(n-1) * g'(x).

Let's find the derivative step by step:

f(x) = (x² + 2x + 1)⁴

f'(x) = 4 * (x² + 2x + 1)³ * (2x + 2)

= 4(x² + 2x + 1)³(2x + 2)

3. To find the derivative of f(x) = 2e^(3x) - 5sin(2x), we can use the rules for differentiating exponential and trigonometric functions.

Let's find the derivative step by step:

f(x) = 2e^(3x) - 5sin(2x)

f'(x) = 2 * d/dx(e^(3x)) - 5 * d/dx(sin(2x))

= 2 * 3e^(3x) - 5 * 2cos(2x)

= 6e^(3x) - 10cos(2x)

4. To find the derivative of f(x) = (2x³+3x-1) / (x²-1), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by (g'(x) * h(x) - g(x) * h'(x)) / (h(x))².

Let's find the derivative step by step:

f(x) = (2x³+3x-1) / (x²-1)

f'(x) = [(6x²+3)(x²-1) - (2x³+3x-1)(2x)] / (x²-1)²

= (6x^4-6x²+3x²-3 - 4x^4 - 6x² + 2x^2 + 6x) / (x²-1)²

= (2x^4 - x² + 3x - 3) / (x²-1)²

5 ?