91Ó°ÊÓ

Implicit differentiation is a technique used when dealing with equations where the dependent variable cannot be explicitly solved for the independent variable. In the context of this exercise, we have an implicit relationship between variables through the composition of functions. The function \(y\) is expressed in terms of another variable \(u\), which in turn depends on \(x\). Thus, we can't directly differentiate \(y\) with respect to \(x\) without first considering how \(u\) behaves with respect to \(x\).
In such scenarios, implicit differentiation allows us to handle the differentiation indirectly by using the chain rule. By expressing the derivative of \(y\) in terms of the derivative of \(u\) and the derivative of \(g(x)\), we obtain the change of \(y\) relative to \(x\). This procedure is especially effective when functions are nested within each other, and direct separation is not possible.

Trigonometric derivatives are essential when working with functions involving trigonometric expressions, like \(\sin\), \(\cos\), and \(\tan\). In this exercise, we differentiated \(y = \sin(u)\) with respect to \(u\). The derivative of \(\sin(u)\) is \(\cos(u)\). This change is based on the fundamental rule that connects the sine and cosine functions through differentiation.

Similarly, during the differentiation of \(u = x - \cos(x)\), the derivative of \(-\cos(x)\) is \(\sin(x)\), accounting for the negative sign. Knowing these basic trigonometric derivatives enables us to handle more complex expressions and execute calculations rapidly. Trigonometric identities and derivatives frequently appear in various calculus problems, making these fundamental skills indispensable.

• Derivative of \(\sin(x)\): \(\cos(x)\)
• Derivative of \(\cos(x)\): \(-\sin(x)\)
• Derivative of \(\tan(x)\): \(\sec^2(x)\)

Composite functions involve one function nested within another, such as \(y = f(g(x))\), a scenario present in this exercise. In dealing with composite functions, it becomes essential to apply the chain rule, which offers a systematic way to differentiate them.
To differentiate a composite function like the one in this exercise, follow these steps:

• First, identify the outer function \(f(u) = \sin(u)\), and differentiate it with respect to the inner function \(u\), giving \(f'(u) = \cos(u)\).
• Next, identify the inner function \(u = g(x)\), which is expressed as \(x - \cos(x)\), and differentiate it with respect to \(x\), resulting in \(g'(x) = 1 + \sin(x)\).
• Finally, apply the chain rule: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).

The chain rule enables you to calculate the derivative of the composite function effectively, linking the rates of change via multiplication. Recognizing when a function is composite and knowing how to apply the chain rule is key to mastering calculus.