sing 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 a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

The chain rule states that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function.

Let's break down the function into two parts: u = 2x - 11 and v = e^(-43x).

The derivative of u, u' = 2 (using the power rule).

The derivative of v, v' = e^(-43x) * -43 (using the chain rule).

Now, apply the product rule: f'(x) = u'v + uv'

f'(x) = 2e^(-43x) - 43(2x - 11)e^(-43x)

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

The derivative of ln(u) is 1/u * u', where u' is the derivative of u.

Let u = 12x^9 - 7. The derivative of u, u' = 108x^8 (using the power rule).

Now, apply the chain rule: f'(x) = 1/u * u'

f'(x) = 108x^8 / (12x^9 - 7)