Using the "f prime" notation can get cumbersome.

There is a different way to symbolize differentiation (the process of taking the derivative of a function). We use this symbol:

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [   \right ] $$

This is read as the derivative with respect to x of.

Using our example,

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [  x^2 + 1  \right ]   =   2 x $$

We can use different variables, too:

$$ {\operatorname{d} \over \operatorname{d}\!q} \left [  q^2 + 1  \right ]   =   2 q $$

For the derivative with respect to x at a given point \( x_0 \), we say:

$$ \left.  {{\operatorname{d} \over \operatorname{d}\!x} \left [  x^2 + 1  \right ] } \right |_{x = x_0}  =   2 x_0 $$

When we have a dependent variable \( y \), as in

$$ y = f(x) $$

we write:

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} \left [    \right ] $$

as in

$$ y = x^2 + 1 $$

whose derivative is

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} \left [  x^2 + 1  \right ]    =   2 x $$

Some more examples of this notation style:

$$ {\operatorname{d}\!y \over \operatorname{d}\!u} \left [  u^2 + 1  \right ]    =   2 u $$

$$ \left . {{\operatorname{d}\!y \over \operatorname{d}\!z} \left [  z^2 + 1  \right ] } \right |_{z = z_0}  =   2 z_0 $$

We denote a change in the value of \( x \) by using the delta notation:

$$ \Delta x = x_1  -  x_0 $$

Such a change in \( x \) produces a corresponding change in \( y \):

$$ \Delta y   =   y_1  -  y_0   =   f(x_1) - f(x_0) $$

We can get rid of \( x_1 \) and \( y_1 \) by using delta notation this way:

$$ \Delta y   =   f( x_0 + \Delta x )  -  f(x_0) $$

or more simply

$$ \Delta y   =   f( x + \Delta x )  -  f(x) $$

With delta notation, we can change our complicated-looking formula for the derivative:

$${\lim_{ d \rightarrow 0 }} {{ f( x_0 + d ) - f( x_0 ) } \over d } $$

to something simpler:

$$ {\operatorname{d}\!y \over \operatorname{d}\!x}   =  { \lim_{ \Delta x \rightarrow 0 } {{ \Delta y } \over { \Delta x } } } $$

The symbols \( \operatorname{d}\!y \) and \( \operatorname{d}\!x \) are called differentials.

We can rewrite

$$ {\operatorname{d}\!y \over \operatorname{d}\!x}   =  { 2 x } $$

as

$$ {\operatorname{d}\!y  = 2 x  \operatorname{d}\!x} $$

This notation is called the differential form.

We say a change of \( \operatorname{d}\!x \) units in \( x \) produces a change of \( \operatorname{d}\!y \) units in \( y \).