Using the Limit Definition of a Derivative to Calculate the Derivative of ( frac{4}{x} - x )

In this article, we will explore how to use the limit definition of a derivative to calculate the derivative of the function ( f(x) frac{4}{x} - x ). The limit definition of a derivative is a fundamental concept in calculus that allows us to find the instantaneous rate of change of a function. This method is crucial for understanding more advanced calculus topics and can be a powerful tool in mathematical analysis.

Understanding the Limit Definition of a Derivative

The limit definition of a derivative of a function ( f(x) ) at a point ( x ) is given by:

[ f'(x) lim_{h to 0} frac{f(x h) - f(x)}{h} ]

Applying the Limit Definition to ( f(x) frac{4}{x} - x )

Let's begin by setting ( f(x) frac{4}{x} - x ). To find the derivative, we need to calculate the difference quotient:

[ frac{f(x h) - f(x)}{h} ]

First, we compute ( f(x h) ):

[ f(x h) frac{4}{x h} - (x h) ]

Now, we substitute ( f(x h) ) and ( f(x) ) into the difference quotient:

[ frac{left( frac{4}{x h} - (x h) right) - left( frac{4}{x} - x right)}{h} ]

Simplify the numerator:

[ frac{frac{4}{x h} - x - h - frac{4}{x} x}{h} ]

Further simplification gives:

[ frac{frac{4}{x h} - frac{4}{x} - h}{h} ]

Combine the fractions in the numerator:

[ frac{frac{4x - 4(x h)}{x(x h)} - h}{h} ]

Simplify the fraction:

[ frac{frac{4x - 4x - 4h}{x(x h)} - h}{h} ]

Which simplifies to:

[ frac{frac{-4h}{x(x h)} - h}{h} ]

Further simplifying:

[ frac{-4h - hx(x h)}{hx(x h)} ]

Which simplifies to:

[ frac{-4 - h(x h)}{x(x h)} ]

As ( h to 0 ) :

[ lim_{h to 0} left( frac{-4}{x(x h)} - 1 right) ]

Since ( h to 0 ) :

[ frac{-4}{x^2} - 1 ]

Conclusion

Therefore, the derivative of ( f(x) frac{4}{x} - x ) is:

[ f'(x) frac{-4}{x^2} - 1 ]

This result confirms the correctness of the limit definition approach and demonstrates the powerful utility of this method in calculus. By applying the limit definition, we gain a firm understanding of how a function changes at each point, which is a cornerstone of differential calculus.

Keywords: limit definition of a derivative, derivative calculation, mathematical functions