Lec 3 | MIT 18.01 Single Variable Calculus, Fall 2007

The paragraph introduces the purpose of the video, which is to develop more derivative formulas to achieve the goal of differentiating any function. It highlights the importance of specific and general derivative formulas, such as those for x^n, (1/x), and the rules for combining functions through addition and multiplication by a constant. The speaker, a professor, emphasizes the utility of these formulas for polynomials and other functions and mentions the focus on trigonometric functions, specifically sine and cosine derivatives, in this lecture. The video is part of MIT OpenCourseWare, a free educational resource supported by donations.

The professor begins the derivation of the sine function's derivative by applying the difference quotient and the sum formula for sine. The process involves taking sin(x + delta x), subtracting sin(x), and dividing by delta x. The sum formula for sine, sin(a + b) = sin(a)cos(b) + cos(a)sin(b), is then used to simplify the expression. The goal is to simplify and group terms in a way that will allow the limit to be taken as delta x approaches zero, which is a crucial step in finding the derivative.

The paragraph delves into the limit process necessary to find the derivative of the sine function. The professor explains the need to group terms so that the limit can be taken without resulting in a 0/0 indeterminate form. By recognizing that cosine of zero is one, the terms are grouped and simplified to reveal that the derivative of sine with respect to x is cosine, as the limit of (sin(delta x) - sin(x)) / delta x approaches zero. This leads to the conclusion that the derivative of sine is cosine, simplified as cos(x).

The professor transitions to the derivative of the cosine function, using the sum rule for cosine to set up the difference quotient. The goal is to simplify the expression in a similar manner to the sine function's derivative, resulting in an expression involving cos(x)cos(delta x) - sin(x)sin(delta x). The paragraph also discusses the structure of the proofs for the derivatives of sine and cosine, highlighting that knowing the rate of change at x = 0 allows for determining the derivatives at all points.

The paragraph presents a geometric proof for the derivative of the cosine function. The proof involves drawing a unit circle and using geometric interpretations of the sine and cosine functions. The professor introduces the concept of replacing delta x with an angle theta and explains how the limit of (cos(theta) - 1) / theta tends to zero as theta approaches zero, which is a key step in proving the derivative of cosine.

The professor provides a geometric interpretation of the derivatives of sine and cosine functions. The explanation involves visualizing the unit circle and the angles involved, with the aim of demonstrating how the derivatives are derived from the geometric properties of the circle. The emphasis is on the concept that short pieces of curves on the unit circle can be approximated as straight lines, which simplifies the calculation of derivatives.

The paragraph involves a Q&A session where the professor addresses students' questions about the proofs for the derivatives of sine and cosine. Topics discussed include the interpretation of the limit as 0/0, the significance of radians in the geometric proofs, and the rationale behind the specific lengths used in the unit circle diagrams. The professor clarifies misconceptions and reinforces the importance of using the correct units for angles.

The final paragraph introduces general rules for differentiation that are applicable to a wide range of functions. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second, plus the first function times the derivative of the second. The quotient rule provides a formula for the derivative of a quotient of two functions, which involves the product of the numerator's derivative and the denominator, minus the product of the numerator and the denominator's derivative, all divided by the square of the denominator. The professor emphasizes the utility of these rules in various calculus applications.