Q.Find the derivative of $f(x)=\dfrac{1}{x}$ ($x\neq0$) from first principles.

The derivative of a function $f$ at a point

$a$ is formally defined as $f'(a)=\lim_{h\to0}[f(a+h)-f(a)]/h$, provided this limit exists, in

which case $f$ is called differentiable at $a$. Computing $f'(x)$ for a general point $x$

directly from this limit -- expanding $f(x+h)$, subtracting $f(x)$, dividing by $h$, and only

then letting $h\to0$ -- is called differentiating from first principles, and is the foundation

every later shortcut formula (the power rule, the product and quotient rules, the trigonometric

derivatives) is built on and justified by. As with ordinary limits, a left-hand derivative and a …