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#