The derivative of 1/u for all u(x) not zero, is given by: -u’/u^2. Let’s prove it using limits.

Derivative of 1/u(x)

Let u(x) be a function of real variable x such that u(x)0, and let f(x)=1u(x).

The derivative f(x) of the function f(x) is: xR,f(x)=u(x)u2(x)

Proof

Let xR be such that u(x)0. f(x)=limh0f(x+h)f(x)h=limh01u(x+h)1u(x)h=limh0u(x)u(x+h)hu(x)u(x+h)=limh0(u(x+h)u(x))hu(x)u(x+h)=limh0(u(x+h)u(x))h1u(x)u(x+h)=limh0u(x)1u(x)u(x+h)=limh0u(x)u(x)u(x+h)=u(x)u2(x)

Therefore, we have: xR,f(x)=u(x)u2(x)