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The chain rule is a fundamental method in calculus used for finding the derivative of composite functions. Composite functions are functions made up of one function inside another, such as \(f(g(x))\). The chain rule can be remembered as follows: If you have a function \(y = g(f(x))\), the derivative \(dy/dx\) is found by multiplying the derivative of the outer function \(g\) with respect to the inner function \(f\), and then the derivative of the inner function \(f\) with respect to \(x\). In mathematical terms: \[\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\] The chain rule is essential for differentiating complex expressions. For example, in differentiating \(p(x) = e^{u(x)}\), consider \(v = u(x)\). The derivative is then \(e^v \cdot u'(x)\) or \(e^{u(x)} \cdot u'(x)\).

A function is differentiable if it has a derivative at every point in its domain. This means the function must be smooth and not have any sharp corners or discontinuities. In more formal terms, a function \(f(x)\) is differentiable at \(x = a\) if the limit: \[\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\] exists. Differentiability implies continuity; if a function is differentiable, it must also be continuous. For instance, in the exercises given, functions like \(u(x)\) and \(e^x\) must be differentiable for us to apply the chain rule effectively. When differentiating \(s(x) = u(\cot(x))\), the existence and smoothness of \(u(x)\) are critical to ensuring that \(u'(x)\) exists and is well-behaved.

The power rule is a basic derivative rule used when differentiating functions of the form \(x^n\), where \(n\) is any real number. The power rule formula is: \[\frac{d}{dx}(x^n) = nx^{n-1}\] This rule is simple but incredibly powerful for differentiating polynomials and other algebraic expressions. For example, in the problem \(a(x) = u(x^4)\), we let \(w = x^4\). The derivative of \(w\) with respect to \(x\) is \(4x^3\), which is derived using the power rule. Therefore, \(a'(x) = u'(x^4) \cdot 4x^3\). The power rule is also integrated with the chain rule for differentiating composite functions involving powers.

The cotangent function, denoted as \(\cot(x)\), is the reciprocal of the tangent function. Mathematically, it is given by: \[\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}\] Differentiating the cotangent function requires the application of the chain rule, especially in composite functions. The derivative of \(\cot(x)\) is: \[\frac{d}{dx} \cot(x) = -\csc^2(x)\] where \(\csc(x) = \frac{1}{\sin(x)}\) is the cosecant function. For the problem \(r(x) = \cot(u(x))\), let \(v = u(x)\). The derivative will be \(-\csc^2(v) \cdot u'(x)\). Furthermore, for \(s(x) = u(\cot(x))\), the chain rule gives the derivative as \(u'(\cot(x)) \cdot (-\csc^2(x))\).

The exponential function is one of the most important functions in calculus and is usually denoted as \(e^x\). It has the unique property where its derivative is the same as the function itself: \[\frac{d}{dx}e^x = e^x\] This property makes the exponential function incredibly straightforward to differentiate. For composite functions involving exponentials, the chain rule is typically applied. For instance, consider the problem \(p(x) = e^{u(x)}\). Here, we set \(v = u(x)\), and the derivative will be \(e^v \cdot u'(x)\), simplified to \(e^{u(x)} \cdot u'(x)\). Similarly, in \(q(x) = u(e^x)\), by letting \(w = e^x\), the derivative becomes \(u'(e^x) \cdot e^x\). This repetitive structure showcases the beautiful simplicity and power of the exponential function in calculus.