Question

If y=1/rtx. Find dy/dx from the first principle.

Answer 1

so, assuming that r and t are constants, the difference quotient is
(f(x+h)-f(x))/h
= (1/(rt(x+h)) - 1/(rtx))/h
= 1/(rt) (1/(x+h) - 1/x)/h * (1/(x+h) + 1/x)/(1/(x+h) + 1/x)
= 1/(rt) (1/(x+h)^2 - 1/x^2)/(h(1/(x+h) + 1/x))
= 1/(rt) (x^2-(x+h)^2)/(h(x+h)^2x^2(1/(x+h) _ 1/x))
= 1/(rt) (-2hx-h^2)/h * 1/((x+h)^2x^2(1/(x+h) + 1/x))
= 1/(rt) (-2x-h)/((x+h)^2x^2(1/(x+h) + 1/x))
now take the limit as h⊥0 and you have
1/(rt) (-2x)/(x^2 x^2(2/x))
= 1/(rt) * -1/x^2
= -1/(rtx^2)