Surely, let's go through your question step by step.
1. Given the function f(x) = e^(x^2 + 2x - 1), the derivative of this function is found using the chain rule for differentiation which states: the derivative of a composite function is the derivative of the outside function evaluated at the inside function times the derivative of the inside function. In this case, we have:
Derivative = (e^(x^2 + 2x - 1))' * (x^2 + 2x - 1)'
Here, the derivative of the exponential function is itself, and the derivative of (x^2 + 2x - 1) is (2x + 2). Thus, the derivative of e^(x^2 + 2x - 1) = (2x + 2)*exp(x^2 + 2x - 1)
2. For the function f(x) = ln(x) / x, we can use the quotient rule, which states that the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Applying this rule, we get:
Derivative = {x*(1/x) - ln(x)*(1)}/x^2 = (1-ln(x))/x^2
Therefore, the derivative of ln(x)/x = -log(x)/x^2 + x^(-2)
3. Given the function f(x) = 4^(-4x^3 + 2x), taking the derivative involves using the general exponential rule and the chain rule again (as in problem 1). Hence, we find:
Derivative = {4^(-4x^3 + 2x)}* log(4) * (-12x^2 + 2) = 4^(-4*x^3 + 2*x) * (2-12*x^2) * log(4)
4. Finally, we consider the function f(x) = log(2x^2 + 2). The derivative here involves the chain rule once more:
Derivative = {(2x^2+2)}^(-1) * (4x) = 4x/(2*x^2 + 2)
And those are the derivatives of your given functions!