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When you deal with derivatives of functions involving division, you will use the quotient rule. It helps calculate the derivative of a ratio of two differentiable functions. The formula for the quotient rule is given as:

• \[\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\]

Here, \( u \) and \( v \) are functions of \( x \). The prime symbol (') denotes the derivative of that function. Start by identifying which is the numerator \( u \) and which is the denominator \( v \).

After identifying them, calculate:\( u' \), the derivative of the numerator, \( v' \), the derivative of the denominator. Substitute them back into the formula for a complete expression of the derivative.

In the exercise above, notice how each term was treated separately using this rule and carefully substitute values to achieve the final answer.

Differentiation is a core concept in calculus that allows you to find the rate at which something changes. When you want to determine the slope of a curve or how functions change at specific points, you will differentiate.

When differentiating functions, remember the basic rules:

• Power rule for polynomials
• Product rule for products of functions
• Quotient rule, as discussed
• Chain rule for composite functions

In the provided solution, the differentiation process is started by breaking down complex expressions. By dealing with each term separately, it's easier to handle. This breakdown helps in managing operations like subtraction and division during differentiation.

The process involves applying appropriate rules based on the structure of each term, simplifying derivatives where possible, and finally putting everything together for the final answer.

Trigonometric functions, like sin and cos, often appear in differentiation problems. These functions are cyclic, and their derivatives also have specific patterns:

• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\tan x\) is \(\sec^2 x\).

Understanding these derivatives assists in handling expressions involving trigonometric components.

With trigonometric functions included, differentiation requires attention to both the function and their derivatives. Make sure to keep track of positive and negative signs as these are crucial in accurate calculation.

In the exercise, using the derivatives of \(\cos x\) and \(\sin x\), and incorporating them correctly in the formulae ensures precision in finding the right derivative. Always simplify your answer when possible, even though sometimes it can't be further simplified.