Fourth derivative of $f(x)=e^{4/x}$

Question

Fourth derivative of $f(x)=e^{4/x}$

I keep trying to calculate this but I can't get it right… Can somebody walk me through this? I get it's just the product rule and chain rule, but dang!! The correct answer is supposed to be:

$$f^{(4)}(x)=\frac{4e^{4/x}(24x^3+144x^2+192x+64)}{x^8}$$

Answer 1

If you log the function you get the first derivative equal to $$ h_1(x) = f'(x)=-\frac{4}{x^2}e^{\frac{4}{x}} $$ The second derivative will be $$ h_2(x) = \frac{8}{x^3}e^{\frac{4}{x}} -\frac{4}{x^2}f'(x) $$ And you can use the first derivative. Repeat twice more.