Lec 5 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins the lecture by expressing gratitude for the support of MIT OpenCourseWare and introduces the topic of implicit differentiation. This technique is highlighted as a powerful method for differentiating functions that are not easily expressed in explicit form. The lecture aims to extend the differentiation capabilities to rational exponents and nth roots using the chain rule in an innovative algebraic manner. The first example involves differentiating y = x^(m/n) by rewriting the equation to y^n = x^m and applying the chain rule, leading to the discovery of the derivative dy/dx in terms of x.

Building on the foundation of implicit differentiation, the professor demonstrates how to handle exponents that are rational numbers, specifically focusing on the case where 'a' is expressed as m/n, with m and n being integers. The process involves rewriting the function to a form that allows the application of known differentiation rules, such as integer powers, and then solving algebraically for dy/dx. The lecture illustrates this with a detailed walkthrough of the steps and the algebraic manipulations required to arrive at the derivative in terms of x.

The professor presents a classic example of implicit differentiation by differentiating the equation x^2 + y^2 = 1, which defines a circle. The process involves differentiating both sides of the equation with respect to x and then solving for the derivative of y with respect to x, denoted as y'. The professor contrasts this implicit method with the explicit method, which involves solving for y first and then differentiating, highlighting the simplicity and directness of the implicit approach.

The lecture continues with a comparison between implicit and explicit differentiation methods. The professor demonstrates that while explicit differentiation can be more complex and requires solving for y in terms of x, implicit differentiation allows for direct differentiation of the given equation. This approach is shown to be more efficient, especially when dealing with complex functions, as it avoids the need for simplification before differentiation.

The professor tackles a more complex example involving a fourth-order equation, y^4 + xy^2 - 2 = 0. The solution process involves using implicit differentiation to find the derivative of y with respect to x, denoted as y'. The professor shows that by differentiating the equation and rearranging terms, a formula for y' can be derived without needing to solve for y explicitly. This method is particularly useful for equations that are difficult to solve explicitly.

The lecture concludes with an exploration of how implicit differentiation can be applied to find the derivatives of inverse functions. The professor uses the example of the inverse tangent function, or arctan(x), to illustrate this. By starting with the defining equation for the inverse function, differentiating both sides, and then solving for the derivative, the professor demonstrates that implicit differentiation provides a powerful tool for finding derivatives of functions that are not easily invertible.

The professor provides a detailed explanation of how to find the derivatives of inverse trigonometric functions, specifically focusing on the inverse tangent and inverse sine. Using implicit differentiation, the professor derives the derivative of arctan(x) as 1 / (1 + x^2) and the derivative of arcsin(x) as 1 / sqrt(1 - x^2). The importance of understanding the range of applicability for these derivatives is emphasized, and the professor warns about the need to be careful with the domain and range of the functions involved.