Given function: f(x)=-2\sqrtx then its derivative using first principle as follows
\frac{d}{dx}f(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
f'(x)=\lim_{h\to 0}\frac{-2\sqrt{x+h}-(-2\sqrtx)}{h}
=\lim_{h\to 0}\frac{-2\sqrtx(\sqrt{\frac{x+h}{x}}-1)}{h}
=-2\sqrtx\lim_{h\to 0}\frac{\sqrt{(1+h/x)}-1}{h}
=-2\sqrtx\lim_{h\to 0}\frac{(1+h/x)^{1/2}-1}{h}
=-2\sqrtx\lim_{h\to 0}\frac{(1+(1/2) h/x+(1/2)(1/2-1)(h/x)^2+\ldots)-1}{h}
=-2\sqrtx\lim_{h\to 0}\frac{(1/2) h/x+(1/2)(-1/2)(h/x)^2+\ldots}{h}
=-2\sqrtx\lim_{h\to 0}((1/2) 1/x+(1/2)(-1/2)h/x^2+\ldots)
=-2\sqrtx((1/2) 1/x+0)
=-1/\sqrtx