\displaystyle 1.\ \text{Let } y=f(x)\ \text{be a function of } x,\ \text{and let } \Delta x\ \text{be a small change in } x \text{ and } \\ \Delta y \text{ be the corresponding change in } y.\ \text{Then, } \Delta y=\frac{dy}{dx}\Delta x\ \text{approximately.}
\displaystyle \frac{dy}{dx}\Delta x\ \text{is called differential of } y \text{ and is denoted by } dy.

\displaystyle 2.\ \text{Following are some useful results on differentials:}
\displaystyle (i)\ \text{If } f(x)\ \text{is a constant function, then its differential is zero.}
\displaystyle (ii)\ \text{If } y=cu,\ \text{then } dy=c\,du,\ c \text{ is a constant.}
\displaystyle (iii)\ \text{If } y=u\pm v,\ \text{then } dy=du\pm dv.
\displaystyle (iv)\ \text{If } y=uv,\ \text{then } dy=u\,dv+v\,du.
\displaystyle (v)\ \text{If } y=\frac{u}{v},\ \text{then } dy=\frac{v\,du-u\,dv}{v^2}.
\displaystyle (vi)\ \text{If } y=f(x),\ \text{then } dy=f'(x)\,dx.

\displaystyle 3.\ (i)\ \text{Let } y=f(x)\ \text{be a given function of } x.\ \text{If } \Delta x \text{ is an error in } x, \text{then the} \\ \text{corresponding error }
\displaystyle \Delta y \text{ in } y \text{ is given by } \Delta y=\frac{dy}{dx}\Delta x.
\displaystyle \text{The error } \Delta x \text{ in } x \text{ and } \Delta y \text{ in } y \text{ are known as absolute errors.}
\displaystyle (ii)\ \text{If } \Delta x \text{ is an error in } x,\ \text{then } \frac{\Delta x}{x} \text{ is called relative error in } x.
\displaystyle (iii)\ \text{If } \Delta x \text{ is an error in } x,\ \text{then } \frac{\Delta x}{x}\times 100 \text{ is called the percentage error in } x.

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