Derivatives of Trigonometric Functions - Product Rule Quotient & Chain Rule - Calculus Tutorial

The video demonstrates how to prove the derivative of sec(x) is sec(x)tan(x). This is done by rewriting sec(x) as 1/cos(x) or cos(x)⁻¹ and applying the power rule and chain rule, then simplifying using trigonometric identities.

This section introduces the derivatives of the six fundamental trigonometric functions: sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x). A helpful tip for memorization is provided: derivatives of co-functions (cosine, cosecant, cotangent) typically have a negative sign.

An example demonstrates finding the derivative of a function combining a polynomial term with sine and cosine functions. The power rule is used for the polynomial term, and the previously learned trigonometric derivatives are applied.

The video explains how to apply the product rule (f'g + fg') when a function is a product of two terms, one of which is a trigonometric function. Examples include x²sin(x) and x³cos(x), detailing the step-by-step process of identifying f, g, and their derivatives.

The quotient rule (g f' - f g' / g²) is introduced for functions involving division. Examples like sin(x)/x² and (1+sin(x))/(x-tan(x)) illustrate how to differentiate such expressions, including simplifying the final result.

This part explains the chain rule for composite functions, f(g(x)), which involves differentiating the outside function while keeping the inside the same, then multiplying by the derivative of the inside function. Specific formulas for derivatives of trigonometric functions with an inner function 'u' are provided.

Several examples demonstrate the application of the chain rule. These include sin(5x), cos(x³), sec(x²), and tan(sin(4x)), progressively showing how to differentiate from the outermost function inwards.

More complex examples involving powers of trigonometric functions are tackled, like sin²(3x) and cot⁴(sin(x³)). The strategy involves rewriting the function to clearly identify the outermost exponent before applying the chain rule iteratively.

The final section proves that the derivative of cot(x) is -csc²(x). This involves rewriting cot(x) as cos(x)/sin(x) and applying the quotient rule, followed by simplification using the Pythagorean identity sin²(x) + cos²(x) = 1.