Using the product rule, quotient rule or chain rule, differentiate each ofthe following functions.i. f (x) = 2x−11e− 43 xii. f (x) = ln (12x9 − 7)

i. To differentiate the function f(x) = 2x−11e^(-43x), we will use the product rule and chain rule.

The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

First, let's identify the two functions in f(x) = 2x−11e^(-43x). We have u(x) = 2x - 11 and v(x) = e^(-43x).

The derivative of u(x) = 2x - 11 is u'(x) = 2.

The derivative of v(x) = e^(-43x) is found using the chain rule. The outer function is e^x and the inner function is -43x. The derivative of e^x is e^x and the derivative of -43x is -43. So, v'(x) = e^(-43x) * -43.

Now, we can apply the product rule:

f'(x) = u(x)v'(x) + v(x)u'(x) = (2x - 11)(-43e^(-43x)) + (e^(-43x))(2) = -86xe^(-43x) + 22e^(-43x) + 4311e^(-43x) = e^(-43x)(-86x + 22 + 473) = -86x*e^(-43x) + 495e^(-43x)

ii. To differentiate the function f(x) = ln(12x^9 - 7), we will use the chain rule.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

The outer function is ln(x) and its derivative is 1/x. The inner function is 12x^9 - 7 and its derivative is 108x^8.

So, the derivative