CHAPTER 5
### Chapter 5 - Differentiating Functions

#### Section 5.1 -
Differentiating
Functions

Differentiation is the process of
finding the rate of change of a function. We have proven that if *f* is a variable dependent on an
independent variable *x*, such that
then
where n is a positive
integer. The derivative reflects the instantaneous rate of change of the
function at any value *x*. The
derivative is also a function of *x*
whose value is **dependent** on *x.*

Take a look at the left side of the
function,
By definition the
derivative of a dependent variable, *f*,
is
, which is the instantaneous rate
of change of
* f *with respect to *x* at any condition *x*. The
right side of the function,
, represents the independent variable whose derivative is

When differentiating a function of
the form
, the derivative of the dependent variable is,
, and the derivative of the independent variable is
. Thus differentiating a function results in a new function
of *x*, where
. The derivative is called
, read “*f* prime of *x*”, and it represents the derivative of
a function of *x* **with respect to the independent
variable, ***x*.. If
, then:

gives the
instantaneous rate of change of *f(x)*
as a function of any value, *x. *Remember
that the rate of change of a function other than a line is not constant. Its
value changes as *x *changes.

If* f(x)* were equal to a constant multiplied by a function of x such
as:

The derivative
of f(x) would be:

Thus the derivative of *f(x)* with respect to *x,* is the constant multiplied by the
derivative of the function of *x, A(x).*

**Next section ->**
*
Section 5.2 - Differentiating Sums of Functions
*