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,**.. If , then:*x*
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).*
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