<s>[INST] How to get the derivative How do I find the derivative of $\displaystyle \frac{1+4\cos x}{2 \sqrt{x+4\sin x}}?$
So I got this as my answer: 
$$\frac{(4 \cos x+1)^2}{4\sqrt{4 \sin x+x}}-2 \sin x \sqrt{4\sin x+x}$$
does this look correct?
 [/INST] Do it in order
$$\frac{\frac{d}{dx}(1+4\cos x)2\sqrt{x+4\sin x} -\frac{d}{dx}(2\sqrt{x+4\sin x})(1+4\cos x)}{4(x+4\sin x)}.$$
Solve the derivatives one by one and then replace in the big fraction. Remember to use the chain rule!
</s>