91Ó°ÊÓ

Derivatives are a fundamental concept in calculus. They represent the rate at which a function is changing at any given point. In simple terms, the derivative of a function gives you the slope of the tangent line at any point on the function's graph.
To find the derivative, you often use different rules depending on the form of the function. In this case, with a function written as a quotient, we use the quotient rule. The quotient rule is expressed as:

• If you have a function \( y = \frac{u(x)}{v(x)} \), then the derivative \( y' \) is: \( y' = \frac{u' \, v - u \, v'}{v^2} \).
• Here, \( u(x) \) and \( v(x) \) are functions of \( x \), and \( u'(x) \) and \( v'(x) \) are their derivatives.

Calculating derivatives with the quotient rule involves these steps:

• First, identify \( u(x) \) and \( v(x) \) from the function.
• Next, find the derivatives \( u'(x) \) and \( v'(x) \).
• Then apply these values to the quotient rule formula.

Trigonometric functions like sine and cosine play an important role in calculus, especially when finding derivatives. These functions model periodic phenomena, such as waves. When dealing with functions that involve these, you need to apply specific rules for differentiation.
For the sine function:- The derivative of \( \, \sin(x) \, \) is \( \, \cos(x) \, \).
For the cosine function:- The derivative of \( \, \cos(x) \, \) is \( \, -\sin(x) \, \).
When you have trigonometric functions with multipliers inside, such as \( \, \sin(2x) \, \) or \( \, \cos(3x) \, \), don't forget to use chain rule, which involves multiplying by the derivative of the inner function.
The example function \( y = \frac{x + \sin(2x)}{2 + \cos(3x)} \) requires you to calculate:

• \( \frac{d}{dx}(\sin 2x) = 2 \cos 2x \)
• \( \frac{d}{dx}(\cos 3x) = -3 \sin 3x \)

This chain rule application is crucial to reach the correct derivative of the function.

Simplifying expressions is a key step after calculating derivatives to ensure the final answer is in its simplest form. In our exercise, after applying the quotient rule, we arrived at a complex expression:\[ y' = \frac{(1 + 2\cos 2x)(2 + \cos 3x) - (x + \sin 2x)(-3\sin 3x)}{(2 + \cos 3x)^2} \]
To simplify:

• Start by expanding the elements in the numerator.
• Add and subtract like terms from the expanded expression.
• Check and ensure that the denominator remains unchanged.

Through simplification, we reduce the expression to a form that is easier to understand and work with. This leads to:\[ y' = \frac{2 + \cos 3x + 4\cos 2x + 2\cos 2x\cos 3x + 3x\sin 3x + 3\sin 2x\sin 3x}{(2 + \cos 3x)^2} \]True mastery of calculus includes developing the ability to simplify expressions effectively, allowing solutions to be neatly packaged and clearly communicated.