derive xe x

Guri Bee

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18-10-2024

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When it comes to calculus, one of the fundamental concepts is the derivative. A common function that arises in various fields of study, particularly in mathematics, physics, and engineering, is the exponential function ( e^x ). In this article, we will explore how to derive the function ( e^x ), the implications of this derivative, and its applications in real-world scenarios.

What is the Derivative of ( e^x )?

The derivative of a function measures how the function's output changes as its input changes. For the exponential function ( e^x ), the derivative can be calculated using the limit definition of a derivative or recognized as a fundamental property of the function.

The Derivative Calculation

The derivative of ( e^x ) with respect to ( x ) is:

[ \frac{d}{dx}(e^x) = e^x ]

This means that the slope of the tangent line to the curve of ( e^x ) at any point is equal to the value of the function at that point.

Proof Using the Limit Definition

To better understand why this is the case, let’s look at the limit definition of the derivative:

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

For ( f(x) = e^x ):

[ f'(x) = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h} ] [ = \lim_{h \to 0} \frac{e^x \cdot e^h - e^x}{h} ] [ = e^x \lim_{h \to 0} \frac{e^h - 1}{h} ]

It is a known limit that:

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

Thus, we have:

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

This shows that ( e^x ) is unique in that its derivative is equal to itself.

Why is This Important?

The fact that the derivative of ( e^x ) is itself has profound implications in various fields:

1. Growth and Decay Models: In natural growth and decay processes, such as population dynamics or radioactive decay, the change in quantity is proportional to the current amount, modeled by the equation ( \frac{dy}{dt} = ky ), where ( k ) is a constant. The solution to this equation is ( y(t) = y_0 e^{kt} ), highlighting the importance of ( e^x ).

2. Financial Mathematics: In compound interest calculations, the formula for continuous compounding involves ( e ). The understanding of its derivative is crucial for financial professionals when modeling investment growth.

3. Differential Equations: Many differential equations can be solved using exponential functions, and understanding their derivatives helps in finding the solutions efficiently.

Practical Examples

To illustrate the application of the derivative of ( e^x ), let's consider a simple example in finance:

Example: Continuous Compounding

Imagine you invest $1,000 at an annual interest rate of 5% compounded continuously. The formula for the amount ( A ) after time ( t ) years is:

[ A(t) = 1000e^{0.05t} ]

To find out how fast your investment is growing after 10 years, we can calculate the derivative:

[ A'(t) = 1000 \cdot 0.05 \cdot e^{0.05t} ] [ = 50e^{0.05t} ]

After 10 years (( t = 10 )):

[ A'(10) = 50e^{0.5} \approx 50 \cdot 1.6487 \approx 82.43 ]

This means that at ( t = 10 ), your investment is growing at approximately $82.43 per year.

Conclusion

The derivative of ( e^x ) being ( e^x ) itself is a remarkable property that not only simplifies calculations in calculus but also plays a crucial role in various applications across science, finance, and engineering.

Understanding this derivative equips students and professionals with the tools necessary for tackling real-world problems efficiently. By grasping the implications of the exponential function and its derivative, one can appreciate the beauty and utility of mathematics in explaining and predicting natural phenomena.

Further Reading

For those interested in deepening their understanding of derivatives and their applications, consider exploring:

• The Fundamentals of Calculus by James Stewart
• Differential Equations and Their Applications by Morris Tenenbaum and Harry Pollard

By building a solid foundation in calculus, you will be better prepared to tackle more complex mathematical and real-world challenges in the future.