91Ó°ÊÓ

When solving complex mathematical problems, it's common to encounter composite functions, which involve functions within functions. In the given exercise, we see this with the term \(\frac{1}{3} \tan ^{-1} \frac{x}{1-x^{2}}\). Here, the inverse tangent function takes another function \(\frac{x}{1-x^{2}}\) as its input.

To understand composite functions, consider a simpler example: If \(f(x) = x^2\) and \(g(x) = x + 3\), the composite function \(f(g(x))\) would be \(f(g(x)) = (x + 3)^2\). In our exercise, the function within the inverse tangent is a quotient, making the understanding of the composition more critical as it imposes domain restrictions – the values that can be put into the function.

In the inverse tangent's case, the function is not defined when the denominator is zero, so any real number can be an input except values that cause the denominator to be zero. Determining the domain of composite functions requires careful consideration of the domains of the individual functions involved.

The concept of function monotonicity refers to the consistent behavior of a function in terms of increasing or decreasing. More technically, a function is said to be monotonic if it's either entirely non-increasing or non-decreasing throughout its domain.

A function \(f(x)\) is monotonically increasing if, for all \(x_1\) and \(x_2\) where \(x_1 < x_2\), the output \(f(x_1) \leq f(x_2)\). Conversely, it's monotonically decreasing if \(f(x_1) \geq f(x_2)\) under the same conditions.

Looking at our exercise, the inverse tangent function, \(\tan^{-1}(x)\), is a prime example of a monotonically increasing function. As \(x\) increases, so does \(\tan^{-1}(x)\) within its range of \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). Monotonicity assures students that the function won't have any unexpected 'turns' and will be predictable in its growth or decline.

The property of differentiability is a cornerstone in calculus, referring to the ability to take the derivative of a function. If a function is differentiable it means that at any point in its domain, we can find the function's rate of change at that point. This is represented by the function's derivative.

In practical terms, if \(f(x)\) is differentiable, the function has a tangent line that smoothly 'touches' the curve at any point \(x\). This also implies that the function is continuous, as a differentiable function cannot have any breaks or gaps.

The exercise involves the inverse tangent function which is differentiable on its entire domain of real numbers because it has a well-defined slope at each point. Differentiating composite functions, like the ones we see in the exercise, typically requires the use of the chain rule, a fundamental technique in differentiation.