If y = (4x^2)ln(x) then find the second derivative of the function with respect to x when x = e^2 (taken from a C3 past paper)

The first thing to recognise is that this function is a product of two functions: namely, 4x^2 and ln(x), thus we must employ the product rule in order to find the solution. As you may recall, the product rule states that when you have a function f(x) = uv, the differential f'(x) = udv + vdu, thus:

we differentiate once, finding that dy/dx = (4x^2)/x + 8xln(x) and simplify to get the expression 4x + 8xln(x)

then differentiate a second time, remembering to once again employ the product rule for the second term in the equation:

d^2y/dx^2 = 4 + (8 + 8ln(x))

now substitute the value of x = e^2 into the equation:

thus d^2y/dx^2 = 12 + 8ln(e^2)

now as we know that the natural logarithm "ln" is the inverse of the exponential function "e", this becomes:

d^2y/dx^2 = 12 + 8(2)

= 28.