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The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. Its application is crucial when dealing with the derivative of combinations of functions like the one in our exercise: \(y = \log(\sin x + \cos x)\).

To apply the chain rule, first identify the outer and inner functions. Here, the outer function is the logarithm, \(\log(u)\), with the inner function being \(u = \sin x + \cos x\).

When using the chain rule, the derivative of the outer function is taken first. According to the general derivative rule for the natural logarithm, this is \(\frac{1}{u}\). Then, multiply it by the derivative of the inner function, \(u'\).

This results in:

• Derivative of the logarithmic function: \(\frac{d}{dx}[\log(u)] = \frac{1}{u}\times u'\)

In our specific example, applying these steps provides us with the derivative \(\frac{d y}{d x} = \frac{1}{\sin x + \cos x} \times (\cos x - \sin x)\).

In summary, the chain rule is an essential differentiation technique allowing us to break down complex expressions into manageable parts.

Trigonometric functions such as \(\sin x\) and \(\cos x\) are integral in calculus, especially when finding derivatives of expressions involving these elements. In our exercise, the expression inside the logarithm, \(\sin x + \cos x\), involves these fundamental trigonometric functions.

To work with these functions, it's helpful to remember:

• Derivative of \(\sin x\) is \(\cos x\).
• Derivative of \(\cos x\) is \(-\sin x\).

When differentiating \(\sin x + \cos x\), apply these basic rules:

\[\frac{d}{dx}[\sin x + \cos x] = \cos x - \sin x\]

Understanding the derivatives of trigonometric functions helps simplify the process, especially with expressions that involve additive trigonometric terms. This clarity is essential for further steps like applying the chain rule.

Differentiation techniques are tools that help simplify finding the derivative of functions. Among them are the common rules such as product rule, quotient rule, and especially relevant here, the chain rule and derivatives of logarithmic and trigonometric functions.

For our logarithmic function \(y = \log(\sin x + \cos x)\), the necessity is to apply these techniques efficiently to find \(\frac{d y}{d x}\). Here's a summary of how to approach such problems:

• Identify composite functions and decide on differentiation rules to apply. In this case, use the chain rule.
• Recognize and apply the basic rules for trigonometric functions to differentiate them correctly.
• Simplify the expression after differentiation, which often involves combining like terms.

The derivative of \(y\) involves these steps leading to \(\frac{d y}{d x} = \frac{\cos x - \sin x}{\sin x + \cos x}\).

Grasping these differentiation techniques not only aids in tackling more complex problems but also in gaining confidence in calculus methodologies.