How to find first derivative by limit definition of $y=\cot(f(x))$ if $f(x)$ is differentiable?

Question

I tried something using limit's definition formula : $$f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}h$$

Need a bit help on this problem ,not product rule or L'Hopital rule are allowed.

Thank you in advance :)

Answer 1

You can use the chain rule, then you'll have $$\lim_{f(x)\to f(x_0)}\frac{\cot(f(x))-\cot(f(x_0))}{f(x)-f(x_0)}\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$$ Remember that $$f'(x)=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=\lim_{h\to 0}\frac{f(x)-f(x_0)}{x-x_0}$$