Deriving the Derivative of e^x Using the Limit Definition

The function e^x is a classic example often discussed in relation to the chain rule and power rule for differentiation. However, it can be shown that e^x can be directly derived using the limit definition of the derivative, which is a fundamental approach. This alternative method provides a nuanced understanding of how the derivative of e^x is determined.

Step-by-Step Derivation

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

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

Applying the Definition to e^x

Let f(x) e^x. We then have:

f'(x) lim_{h to 0} frac{e^{x h} - e^x}{h}

This expression can be simplified using the property of exponents:

e^{x h} e^x cdot e^h

Substituting this, we obtain:

f'(x) lim_{h to 0} frac{e^x cdot e^h - e^x}{h}

Factoring Out e^x

Factoring e^x out of the limit:

f'(x) e^x lim_{h to 0} frac{e^h - 1}{h}

Evaluating the Limit

The limit lim_{h to 0} frac{e^h - 1}{h} is a well-known limit that equals 1. This can be derived using the series expansion of e^h or L'Hopital's rule. Thus:

lim_{h to 0} frac{e^h - 1}{h} 1

Final Result

Substituting this back into our expression for f'(x) gives:

f'(x) e^x cdot 1 e^x

Conclusion

This approach, while unconventional in its application, shows definitively that the derivative of e^x is e^x. This method serves not only as a rigorous explanation but also as a valuable exploration of the underlying principles of the derivative.

Further Exploration

While the method described above uses the limit definition of the derivative, many derive the derivative of e^x using the chain rule. The chain rule is typically used when the function is a composition of two or more functions.

For example: If g(x) e^u and u f(x), then using the chain rule, the derivative of g(x) with respect to x is:

g'(x) e^u cdot f'(x)

Power Rule

The power rule is used for functions of the form x^n, where n is a constant. It states that the derivative of x^n is n x^{n-1}. However, e^x is not a polynomial and thus does not fit the criteria for direct application of the power rule.

Moreover, the exponential function is a fundamental constant and has unique properties when it comes to differentiation, as demonstrated by the limit definition approach.