91Ó°ÊÓ

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. Suppose you have a function expressed as a fraction, where the numerator is one function, and the denominator is another. In this exercise, that function is \[ u = \frac{\sin x}{1 - \cos x} \].The quotient rule states that if \( f(x) \) and \( g(x) \) are differentiable, then the derivative of their quotient is given by:\[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \]
Applying the quotient rule involves:

• Finding \( f'(x) \), the derivative of the numerator function \( f(x) \).
• Finding \( g'(x) \), the derivative of the denominator function \( g(x) \).
• Substituting these derivatives, along with the original functions, into the quotient rule formula.

Remember, the quotient rule is particularly useful when dealing with ratios involving functions that are not easily separable or multiplicative.

Understanding the derivatives of trigonometric functions is essential when working with calculus problems that involve sinusoidal functions. In our exercise, we deal with \( \sin x \) and \( \cos x \). Knowing their derivatives simplifies our problem greatly.Consider these fundamental derivatives:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).

In our exercise, we differentiate \( f(x) = \sin x \) to get \( f'(x) = \cos x \) and differentiate \( g(x) = 1 - \cos x \) to get \( g'(x) = \sin x \). These derivatives play a crucial role in applying the quotient rule.To master working with trigonometric derivatives, always remember these basic results and use them as building blocks.
Furthermore, don't hesitate to refresh your knowledge of other trigonometric derivatives like those of \( \tan x \), \( \sec x \), and others, as they often interconnect in complex problems.

After applying differentiation rules, such as the quotient rule, simplification of the resulting expression is often necessary. In our exercise, after using the quotient rule on \( \frac{\sin x}{1 - \cos x} \), we obtained:\[ \frac{\cos x - \cos^2 x - \sin^2 x}{(1 - \cos x)^2} \].Simplifying this involves using trigonometric identities, such as \( \sin^2 x + \cos^2 x = 1 \), to combine and reduce terms.

• Substitute known identities to replace and eliminate redundant terms, as seen with \( \sin^2 x = 1 - \cos^2 x \).
• Ensure that the expression is fully simplified to ease understanding and to provide a cleaner result.

In this example, the expression reduces to:\[ \frac{2\sin^2 x - \cos x + 1}{(1 - \cos x)^2} \],Always simplify to the extent possible to make calculations easier and to present a more elegant solution. Practice identifying and employing identities until they become intuitive, as this will aid tremendously in various calculus problems.