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In evaluating the differentiability of functions, we first need to understand the concept of discontinuity. Discontinuity occurs when there is a 'break' or 'gap' in a function. These disruptions can take various forms like jumps, holes, or vertical asymptotes. In the case of the function \(f(x)=\frac{1}{x+1}\), the function presents a discontinuity at \(x=-1\), as the function becomes undefined due to a division by zero. This point of discontinuity is crucial because it is the exact spot where the function fails to be differentiable. As we uncover the behavior of functions, it's necessary to identify these points of discontinuity because they distinguish intervals where a function is smooth and continuous from those where the function is not.

The derivative of a function gives us the slope of the tangent line at any point along the function's graph. When we set out to find derivatives, we are essentially exploring the rate of change of the function's output with respect to its input. For the function \(f(x)=\frac{1}{x+1}\), the derivative, calculated through the power rule, is \(f'(x)=-\frac{1}{(x+1)^2}\). This derivative informs us about the function's rate of change except at the point \(x=-1\), where the denominator of the derivative also becomes zero, hinting at a vertical tangent line, which is not permissible. Thus, the function isn't differentiable at this point. Derivatives also play a key role in many practical applications like physics, economics, and even in the optimization of functions within mathematics.

The tangent line to a function at a given point represents the instantaneous rate of change of the function at that specific spot. This line touches the curve of the function at one point and has the same slope as the curve at that point. For differentiable functions, tangent lines can be found easily for every point on their curves. However, if a function has a discontinuity, as we see with our given function \(f(x)=\frac{1}{x+1}\) at \(x=-1\), it is impossible to draw a tangent line at that point, which indicates the absence of differentiability. Understanding where these tangent lines cannot be drawn is as important as finding the slopes where they exist because it helps to delineate the limitations and behavior of the function's graph.