Question

Differentiate the following functions. $$ u=\frac{\sin x}{1-\cos x} $$

Short answer

Question: Find the derivative of the function \(u = \frac{\sin x}{1 - \cos x}\) with respect to \(x\). Answer: The derivative of the function \(u\) with respect to \(x\) is \(u' = \frac{2\sin^2 x - \cos x + 1}{(1 - \cos x)^2}\).

Step-by-step solution

1. Find the derivatives of f(x) and g(x)

Calculate the derivatives of \(f(x) = \sin x\) and \(g(x) = 1 - \cos x\) with respect to \(x\). For this, we know the following basic derivatives: $$(\sin x)' = \cos x$$ and $$(\cos x)' = -\sin x$$ So the derivatives are: $$f'(x) = (\sin x)' = \cos x$$ and $$g'(x) = (1 - \cos x)' = -(\cos x)' = \sin x$$

2. Apply the quotient rule

Use the quotient rule formula given earlier with the values of \(f(x)\), \(g(x)\), \(f'(x)\) and \(g'(x)\), calculated in step 1: $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$ Replace \(f(x)\), \(g(x)\), \(f'(x)\), and \(g'(x)\) with their respective expressions: $$\frac{d}{dx}\left(\frac{\sin x}{1 - \cos x}\right) = \frac{\cos x (1 - \cos x) - \sin x \sin x}{(1 - \cos x)^2}$$

3. Simplify the expression

Simplify the expression for the derivative by expanding the terms in the numerator and combining similar terms: $$\frac{\cos x - \cos^2 x - \sin^2 x}{(1 - \cos x)^2}$$ Recall the trigonometric identity \(\sin^2 x + \cos^2 x = 1\). This can be used to simplify our expression further, as \(\sin^2 x = 1 - \cos^2 x\). Substitute this into the expression: $$\frac{\cos x - (1- \sin^2 x) - \sin^2 x}{(1 - \cos x)^2}$$ Now, simplify by combining the terms: $$\frac{2\sin^2 x - \cos x + 1}{(1 - \cos x)^2}$$ Thus, the derivative of the given function is: $$u' = \frac{2\sin^2 x - \cos x + 1}{(1 - \cos x)^2}$$

Key concepts

Quotient Rule

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.

Trigonometric Derivatives

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.

Simplifying Expressions

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.