91Ó°ÊÓ

Derivatives express the rate of change of one variable with respect to another. It is a core concept in calculus, measuring how a function value changes as its input changes. In the exercise, we aim to find the derivative using implicit differentiation, which allows us to compute the derivative of functions defined in terms of more than one variable. This method is especially useful when the equation expressing the function is not easily rearrangable to directly apply the regular differentiation rules. To solve this, we typically start with the differentiation of both sides of the equation with respect to one variable, say \( x \).

• Step 1: Differentiate both sides with respect to \( x \). For \( y = \sin x \), the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} \), and for \( \sin x \), it is \( \cos x \). Thus, \( \frac{dy}{dx} = \cos x \).

This allows us to find the derivative with respect to \( y \) by using inverted operations, leading us into the next concepts.

Trigonometric functions, such as \( \sin x \) and \( \cos x \), are periodic functions essential in modeling cyclic phenomena. In calculus, they frequently appear when differentiating functions involving angles. When we differentiate \( \sin x \), we find that its derivative is \( \cos x \). This knowledge is crucial in applying implicit differentiation steps effectively.

• \( \sin x \): Differentiates to \( \cos x \)

Apart from their practical uses, these derivatives are significant in understanding the behavior of wave-like functions in physics and engineering.

The reciprocal rule is applied when we wish to find the derivative of one variable with respect to another by "flipping" the derivative. In our exercise, we started with \( \frac{dy}{dx} = \cos x \) and needed \( \frac{dx}{dy} \). Here, the reciprocal rule directly gives:

• \( \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \)
• Substituting, \( \frac{dx}{dy} = \frac{1}{\cos x} \)

This method is beneficial when inverting derivatives, allowing us to easily bridge between different rates of change. Implicit differentiation relies heavily on this comprehension, making it manageable to switch perspectives or variables frequently.