Lec 6 | MIT 18.01 Single Variable Calculus, Fall 2007

The video script begins with an introduction to the concepts of exponentials and logarithms, which are essential for understanding calculus. The professor emphasizes the importance of these functions, comparing them to trigonometric functions. The base 'a' of an exponential function is introduced, along with its properties, such as any positive number raised to the power of 0 equals 1. The script explains the rule for multiplying exponential functions and how to understand the range of values for a^x, including irrational exponents. The professor also illustrates the exponential function graphically, using y = 2^x as an example, and discusses the rapid growth of exponential functions.

The script then delves into the calculus aspect of exponential functions, specifically focusing on finding their derivatives. The derivative is defined using the limit process, and the professor simplifies the expression for the derivative of a^x by factoring out a^x from the limit. A conceptual shift is introduced where 'delta x' becomes the variable while 'x' is held constant. This leads to the identification of a mystery number M(a), which is defined as the limit of (a^delta x - 1) / delta x as delta x approaches 0. The professor discusses the geometric interpretation of M(a) and its significance in determining the slope of the exponential function at x = 0.

The professor introduces a new concept, the base 'e', defined as the unique number for which the mystery number M(e) equals 1. This leads to the fundamental derivative formula d/dx of e^x being simply e^x. The script explores the implications of this definition, showing how knowing the slope of the graph of e^x at x = 0 allows us to determine the slope at any point. The professor also discusses the process of stretching the function f(x) = 2^x by a factor k, resulting in a new base b = 2^k, and how this relates to finding the base 'e'.

The script introduces the natural logarithm, denoted as ln(x), and its properties derived from the properties of exponential functions. The professor explains the logarithm of a product rule, the logarithm of 1 being 0, and the logarithm of e being 1. These properties are illustrated graphically, showing the inverse relationship between the exponential function e^x and the natural logarithm ln(x). The graph of ln(x) is only defined for positive x, reflecting that e^x is always positive, and the slope of the tangent line to ln(x) at x = 1 is 1.

The script explains how to find the derivative of a logarithm function using implicit differentiation. The process involves rewriting the logarithmic equation as an exponential equation, e^(ln x) = x, and then differentiating both sides with respect to x. The chain rule is applied, leading to the conclusion that the derivative of ln x with respect to x is 1/x. This is a key formula for understanding the rate of change of logarithmic functions.

The script presents a method for differentiating any exponential function a^x by converting it to base e. This involves expressing a^x as e^(x ln a) and then differentiating using the chain rule. The result is that the derivative of a^x is (ln a) a^x. The professor emphasizes the importance of understanding constants in the chain rule and how they affect the rate of change. This method simplifies the process of differentiating exponential functions.

The script introduces logarithmic differentiation, a technique useful when differentiating functions with moving exponents or bases is complex. The method involves taking the logarithm of the function, differentiating the resulting expression, and then using the chain rule to find the derivative of the original function. The professor illustrates this with the function u = x^x, showing how to apply logarithmic differentiation to find its derivative.

The script discusses an example that goes beyond basic calculus, using the tools of logarithmic differentiation to evaluate the limit as n approaches infinity of (1 + 1/n)^n. The professor takes the natural logarithm of the expression, rewrites it in a form that fits the pattern for calculating the derivative of the log function, and shows that the limit of the logarithm is 1. This implies that the original limit is e, providing a numerical way to approximate the base 'e'.

The final paragraph wraps up the discussion by highlighting the significance of the limit evaluation in understanding the base 'e'. The professor points out that this provides a practical numerical method to approximate 'e', such as using (1 + 1/100)^100. This closes the loop on the coherent family of functions discussed in the script, which are well-defined and have practical methods for calculation.