Question

So i am having some issues taking the second derivative of this function, Is anyone able to help? Ive attached the photo in the post below! Thanks in advance!

Answer 1

Here is the equation, Im stuck!

Answer 2

You'd have to use the Product Rule to start with.

f(t) = (t^12)(ln t)
Since... f(t) = a(t) * b(t) when [a(t) = t^12] and [b(t) = ln t]
f'(t) = a'(t) * b(t) + a(t) * b'(t) or...
f'(t) = (12t^11) * (ln t) + (t^12) * (1/t)
f'(t) = (12t^11)(ln t) + (t^11)
f'(t) = (t^11)((12*ln t) + 1)

Then same thing for the second derivative: Product Rule.

f'(t) = (t^11)((12*ln t) + 1)
f"(t) = (11t^10) * ((12*ln t) + 1) + (t^11) * (12/t)
f"(t) = (11t^10)((12*ln t) + 1) + (12t^10)
f"(t) = (t^10)((11(12*ln t) + 1) + 12)

So, in conclusion, when f(t) = (t^12)(ln t)
f'(t) = (t^11)((12*ln t) + 1)
f"(t) = (t^10)((11(12*ln t) + 1) + 12)

Hope this helps!