sing the product rule, quotient rule or chain rule, differentiate each ofthe following functions.i. f (x) = 6x19e−4xii. f (x) = ln(3 x9 + 4)

i. To differentiate the function f(x) = 6x^19 * e^-4x, we will use the product rule which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first.

Let's denote: u = 6x^19 v = e^-4x

Then, we find the derivatives of u and v: u' = 19 * 6x^18 = 114x^18 v' = -4e^-4x

Now, we apply the product rule: f'(x) = u'v + uv' f'(x) = 114x^18 * e^-4x + 6x^19 * -4e^-4x f'(x) = 114x^18e^-4x - 24x^19e^-4x

ii. To differentiate the function f(x) = ln(3x^9 + 4), we will use the chain rule which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Let's denote: u = 3x^9 + 4

Then, we find the derivative of u: u' = 27x^8

Now, we apply the chain rule: f'(x) = 1/u * u' f'(x) = 1/(3x^9 + 4) * 27x^8 f'(x) = 27x^8 / (3x^9 + 4)