Derivative of $\cos(x)^{1/x}$

Question

I'm trying to find the derivative of $$f(x) = \cos(x)^{1/x}.$$

Would it be a good idea to see it as a composition of functions to use the chain rule? Not seeing a clear path to a solution.

Answer 1

$$\left((\cos{x})^{\frac{1}{x}}\right)'=\left(e^{\frac{\ln\cos{x}}{x}}\right)'=e^{\frac{\ln\cos{x}}{x}}\left(\frac{\ln\cos{x}}{x}\right)'=(\cos{x})^{\frac{1}{x}}\left(-\frac{\sin{x}}{x\cos{x}}-\frac{\ln\cos{x}}{x^2}\right).$$

Answer 2

Let $$y=\cos(x)^{1/x}$$ so $$\ln(y)=\frac{1}{x}\ln(\cos(x))$$ using the chain rule we get $$\frac{1}{y}y'=-\frac{1}{x^2}\ln(\cos(x))+\frac{1}{x}\left(\frac{-\sin(x)}{\cos(x)}\right)$$