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

Let's denote u = 6x^19 and v = e^(-4x).

The derivative of u with respect to x is du/dx = 114x^18 (using the power rule).

The derivative of v with respect to x is dv/dx = -4e^(-4x) (using the chain rule).

So, the derivative of f(x) is:

f'(x) = du/dx * v + u * dv/dx = 114x^18 * e^(-4x) + 6x^19 * -4e^(-4x) = 114x^18 * e^(-4x) - 24x^19 * e^(-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.

The derivative of ln(u) with respect to u is 1/u.

The derivative of u with respect to x is du/dx = 27x^8 (using the power rule).

So, the derivative of f(x) is:

f'(x) = d[ln(u)]/du * du/dx = 1/(3x^9 + 4) * 27x^8 = 27x^8 / (3x^9 + 4)