Deriving the Derivative of f(x) 1/x from First Principles

In calculus, one of the fundamental concepts is the idea of a derivative, which describes the rate at which a function changes. Using the first principle of derivatives, or the limit definition of the derivative, we can derive the derivative of any function from scratch.

Introduction to Derivatives

A derivative of a function, ( f(x) ), at a point ( x ) is given by the limit:

f ′ ( x ) lim ( ? → 0 ) f ( x ? ) ? f ( x ) ?

This formula represents the slope of the tangent line to the function at the point ( x ).

Differentiating f(x) 1/x Using the First Principle

Let's consider the function ( f(x) frac{1}{x} ). To find its derivative using the first principle, we follow the steps below:

Start with the limit definition of the derivative: Substitute ( f(x h) ) and ( f(x) ) into the limit definition:

f ′ ( x ) lim ( ? → 0 ) f ( x ? ) ? f ( x ) ?

Substitute ( f(x) frac{1}{x} ) and ( f(x h) frac{1}{x h} ) into the formula:

f ′ ( x ) lim ( ? → 0 ) 1 x ? ? 1 x ?

Combine the fractions in the numerator:

f ′ ( x ) lim ( ? → 0 ) x ? x ? ? x ? x ? ? lim ( ? → 0 ) ? ? x ? x ? ?

Simplify the expression and evaluate the limit:

f ′ ( x ) lim ( ? → 0 ) ? ? ? ? x ? x ? ? ? 1 x ? x ? 1 x ^ 2

Thus, the derivative of ( f(x) frac{1}{x} ) is:

f ′ ( x ) ? 1 x ^ 2

Conclusion

This step-by-step derivation using the first principle provides a clear and detailed approach to understanding the differentiation of ( f(x) frac{1}{x} ). The process involves combining fractions, applying limit concepts, and simplifying expressions, ultimately leading to a clear result.