How to Prove the Derivative of Exponential Functions: ex

In calculus, the derivative of the exponential function ex is a fundamental concept. To illustrate how this can be proven, we will use the limit definition of the derivative and some key properties of the exponential function.

Using the Limit Definition of the Derivative

The derivative of a function f(x) at a point x is defined as:

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

For the function f(x) ex, we need to show that f'(x) ex. Let's proceed step by step using the limit definition:

Start with the limit definition of the derivative:

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

Use the property of exponents am n am an to rewrite the function:

[ lim_{{h to 0}} frac{e^x e^h - e^x}{h} ]

Factor out ex:

[ lim_{{h to 0}} frac{e^x (e^h - 1)}{h} ]

Since ex is constant with respect to h, it can be taken out of the limit:

[ e^x lim_{{h to 0}} frac{e^h - 1}{h} ]

Now, evaluate the limit (lim_{{h to 0}} frac{e^h - 1}{h} 1). This limit is well-known and can be proven using the series expansion of eh:

[ e^h 1 h frac{h^2}{2!} frac{h^3}{3!} ldots ]

Thus:

[ e^h - 1 h frac{h^2}{2!} frac{h^3}{3!} ldots ]

Dividing by h:

[ frac{e^h - 1}{h} 1 frac{h}{2!} frac{h^2}{3!} ldots ]

This expression approaches 1 as h approaches 0.

Substituting back into the derivative expression:

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

Thus, we have proven that the derivative of ex is ex.

Using the Limit Definition with the Base of Natural Logarithm

We can also prove the derivative of ex using the limit definition and the definition of the base of the natural logarithm.

Start with the limit definition of the derivative:

[ f'(z) lim_{{Δz to 0}} frac{e^{z Δz} - e^z}{Δz} ]

Factor an ez out of the fraction:

[ lim_{{Δz to 0}} e^z frac{e^{Δz} - 1}{Δz} ]

Use a property of limits to simplify:

[ e^z lim_{{Δz to 0}} frac{e^{Δz} - 1}{Δz} ]

Recall that the definition of e as the limit of (left(1 frac{1}{n}right)^n) as n → ∞. Substitute (frac{1}{Δz}) for n:

[ e lim_{{Δz to 0}} left(1 frac{1}{Δz}right)^{Δz} ]

Plug this into the limit expression:

[ f'(z) e^z lim_{{Δz to 0}} frac{1 frac{1}{Δz}}{Δz}^{Δz} - 1}{Δz} ]

Simplify with the exponent law:

[ e^z lim_{{Δz to 0}} frac{1 Δz - 1}{Δz} ]

Cancel the ones in the numerator:

[ e^z lim_{{Δz to 0}} frac{Δz}{Δz} ]

Simplify (frac{Δz}{Δz}) to 1:

[ e^z lim_{{Δz to 0}} 1 ]

The limit evaluates to 1:

[ f'(z) e^z cdot 1 e^z ]

Thus, we have proven once again that the derivative of ex is ex.

Conclusion: Using both limit definitions and the properties of the exponential function, we have shown that the derivative of ex is ex. This is a crucial result in calculus and forms the basis for understanding more complex functions involving exponents.