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A differentiable function is one where the derivative exists at each point in its domain. Put simply, if a function is smooth at every point (no sharp corners or cusps), it is differentiable. Differentiability is a crucial property because it allows us to examine how functions change at an infinitesimally small level. Consider the function \( f(x) \). If \( f \) is differentiable at \( x=a \), it means we can compute its derivative, \( f'(a) \). This derivative provides the slope of the tangent line to the curve at that point.

• Function is smooth: No jumps, holes, or sharp corners.
• Derivative exists at each point: We can calculate how the function is changing.
• Local linear approximation: You can approximate the function near any point by a straight line.

Understanding these characteristics is vital as they form the foundation for more advanced concepts like the Inverse Function Theorem, which we’ll dive into next.

The derivative of an inverse function is a fascinating concept. If a function \( f \) is invertible and differentiable, then its inverse \( g = f^{-1} \) is differentiable wherever \( f \) has a non-zero derivative. This is due to the Inverse Function Theorem. Now, how do we find \( g'(y) \)? The theorem gives us a convenient way: \( g'(y) = \frac{1}{f'(x)} \), where \( y = f(x) \). This formula informs us that to find the rate of change of \( g \) at \( y \), we need to take the reciprocal of the rate of change of \( f \) at \( x \).

• Inverse function \( g \): Represents the inverse of \( f \).
• Reciprocal of derivative: \( g'(y) = \frac{1}{f'(x)} \).
• Condition: \( f'(x) eq 0 \) ensures \( g(y) \) is differentiable.

This aspect uncovers how closely related the differentiability and invertibility of \( f \) and \( g \) are.

An inverse function essentially undoes the action of the original function. If you think of \( f(x) \) as a process that takes an input \( x \) and spits out \( y \), the inverse function \( g(y) = f^{-1}(y) \) takes the output \( y \) and returns the original input \( x \). For a function to have an inverse, it must be bijective:

• One-to-One (Injective): Different inputs map to different outputs.
• Onto (Surjective): Every possible output is covered.

These properties ensure that every \( y \) has a unique \( x \) such that \( y = f(x) \).In practice, having a differentiable \( f \) that's also invertible assures us that its inverse \( g \) can be smoothly defined at points where needed. Going back to our exercise, we concluded that if \( f(1) = 8 \) with \( f'(1) = \frac{1}{8} \), the derivative of the inverse at \( y=8 \) was beautifully expressed as \( g'(8) = 8 \), illustrating the harmony between function and inverse dynamics.