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The chain rule is a fundamental technique in calculus used to differentiate compositions of functions.
It states that if you have a function composed of two functions, say \(f(g(x))\), the derivative is found by taking the derivative of the outer function \(f\) evaluated at the inner function \(g(x)\) and then multiplying it by the derivative of the inner function \( g(x) \).
Mathematically, it is written as: \[ (f \,\circ\, g)'(x) = f'(g(x)) \cdot g'(x) \] In the exercise: \[ \sin^{\circ}(x) = \sin \left( \frac{\pi x}{180} \right) \] Let’s set the inner function as \(u = \frac{\pi x}{180}\). By differentiating \( \sin(u) \), we get \( \cos(u) \), and differentiating \( \frac{\pi x}{180} \) gives \( \frac{\pi}{180} \). Putting it together using the chain rule: \[ (\sin^{\circ})'(x) = \cos \left( \frac{\pi x}{180} \right) \cdot \frac{\pi}{180} = \frac{\pi}{180} \cos^{\circ}(x) \] Similarly, differentiating \( \cos^{\circ}(x) \) involves the same steps. The derivative of the outer function \( \cos(u) \) is \( -\sin(u) \): \[ (\cos^{\circ})'(x) = -\sin \left( \frac{\pi x}{180} \right) \cdot \frac{\pi}{180} = -\frac{\pi}{180} \sin^{\circ}(x) \]

Limits help us understand the behavior of functions at specific points or as values approach infinity.
For the exercise, we examine two limits: When \( x\) approaches 0 and when \( x\) approaches infinity. \[ \lim_{x \rightarrow 0} \frac{\sin^{\circ}(x)}{x} \] To solve this, substitute \( \sin^{\circ}(x) \) with \( \sin\left( \frac{\pi x}{180} \right) \). As \( x \rightarrow 0 \), we simplify: \[ \lim_{x \rightarrow 0} \frac{\sin \left( \frac{\pi x}{180} \right)}{x} \] Change variables: \( u = \frac{\pi x}{180} \), so as \( x \rightarrow 0, u \rightarrow 0 \): \[ \lim_{x \rightarrow 0} \frac{\sin^{\circ}(x)}{x} = \lim_{u \rightarrow 0} \frac{\sin(u)}{u} \cdot \frac{180}{\pi} = 1 \cdot \frac{180}{\pi} = \frac{180}{\pi} \] The next limit to solve is \[ \lim_{x \rightarrow \infty} x \sin^{\circ} \left( \frac{1}{x} \right) \] Again, substituting \( \sin^{\circ}(\frac{1}{x}) = \sin\left( \frac{\pi}{180 x} \right) \): \[ \lim_{x \rightarrow \infty} x \sin\left( \frac{\pi}{180 x} \right) \] Change variable: Let \( u = \frac{1}{x} \), so as \( x \rightarrow \infty, u \rightarrow 0 \): \[ \frac{1}{u} \sin(u) \approx \frac{1}{u} \cdot u = 1 \] Thus, the limit is: \[ \lim_{x \rightarrow \infty} x \sin\left( \frac{\pi}{180 x} \right) = \frac{180}{\pi} \]

Differentiation is the process of finding the derivative of a function, which represents the rate of change.
It’s akin to finding the slope of the tangent line at any point on a function's curve. For basic functions like polynomials, the power rule efficiently finds the derivative.
For compositions of functions, like \( \sin^{\circ}(x) \) and \( \cos^{\circ}(x) \), the chain rule is necessary. In differentiation:
• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
The given functions \( \sin^{\circ}(x) = \sin\left( \frac{\pi x}{180} \right) \) and \( \cos^{\circ}(x) = \cos\left( \frac{\pi x}{180}\right) \)
are a composite of trigonometric and linear functions, requiring the chain rule for differentiation.
By letting \( u = \frac{\pi x}{180} \) and following through with the chain rule, derivatives are computed efficiently. In conclusion, these differentiation techniques are foundational tools in calculus, instrumental for solving complex functions and understanding instant rates of change.