Question

How did derive exponential function

Answer 1

Answer: The exponential function is typically defined as:

f(x) = e^x

where e is a mathematical constant approximately equal to 2.71828.

To understand how this function is derived, we need to start with the concept of compound interest. Suppose we have an investment that earns an annual interest rate of r, and let P be the initial principal investment. After one year, the investment will be worth P(1 + r). If we leave the investment for another year, the value will be P(1 + r)^2, and after n years, the value will be P(1 + r)^n.

Now, suppose we want to know what happens to the value of the investment as the time period becomes infinitely small (i.e., in the limit as n approaches infinity). We can do this by considering the limit of the expression (1 + r/n)^n as n approaches infinity. This limit is known as the exponential constant e.

Using calculus, we can prove that the limit of (1 + 1/n)^n as n approaches infinity is equal to e. We can then use this result to define the exponential function as e^x, where x is any real number.

Using the definition of e^x, we can show that the derivative of the exponential function is itself. That is:

d/dx e^x = e^x

This result is known as the exponential function's own derivative property, and it is one of the reasons why the exponential function is so important in mathematics.

Step-by-step explanation:

Answer 2

To derive an exponential function, one can use the relationship that the natural logarithm and the exponential function are inverses, where ln(e^x) = x and e^(ln(x)) = x. This relationship can transform power equations into exponential form using base e and makes multiplication of exponentials a matter of adding exponents.

Step-by-step explanation:

To derive an exponential function, we rely on a critical mathematical relationship between exponential functions and logarithms, specifically the natural logarithm. The natural logarithm ln and the exponential function ex are inverse functions. This means that ln(e^x) = x and e^ln(x) = x.

When working with exponential functions, we often utilize properties such as the rule for multiplication of exponentials: to multiply two exponentials with the same base, you add their exponents. This is why, after n doubling times in an exponential growth scenario, we have an increase by a factor of 2n. For example, after 5 doubling intervals, it would be 25 = 32.

By understanding these properties, we can express any exponential growth equation, for instance, bn as en ln(b). Here, the natural logarithm is employed to convert the base into a form that uses the constant e (approximately 2.71828), which is a fundamental base for exponential functions in mathematics.

This relationship between exponentials and logarithms is crucial, not just for math but for various scientific disciplines. For example, it helps linearize nonlinear relationships such as the exponential rise in vapor pressure with temperature, allowing complex phenomena to be understood and analyzed with greater ease.