91Ó°ÊÓ

To understand derivative calculation, let's initially consider the function we're dealing with: \( f(s) = \sqrt{s} \). Derivatives help us understand how a function changes at any given point. For the function \( f(s) \), we apply the basic rules of differentiation.
For \( f(s) = s^{1/2} \), the derivative, denoted as \( f'(s) \), is calculated using the power rule for derivatives, which states if \( g(s) = s^n \), then \( g'(s) = n \cdot s^{n-1} \). Applying this, we get \( f'(s) = \frac{1}{2} s^{-1/2} = \frac{1}{2\sqrt{s}} \).
Calculating at \( s = 9 \), the derivative \( f'(9) \) gives us \( \frac{1}{2\sqrt{9}} \), which simplifies to \( \frac{1}{6} \). This result indicates the rate of change or the slope of the tangent line at \( s = 9 \) for the function \( f(s) = \sqrt{s} \).

The equation of a tangent line provides a linear approximation of a curve at a particular point. We use the point-slope formula to determine this equation. The point-slope formula is:
\[ y - y_1 = m(x - x_1) \]
Where \( m \) represents the slope and \( (x_1, y_1) \) is a point on the line. In our scenario, the point is \((3, 9)\) since \( f(9) = 3 \) for \( c = 9 \).
We previously determined the slope \( m \) of the tangent line to the inverse function \( f^{-1} \) as \( 6 \). Thus, the equation becomes:
\[ y - 9 = 6(x - 3) \]
This equation can be simplified by distributing and combining like terms:
\[ y = 6x - 18 + 9 \]
Which results in:
\[ y = 6x - 9 \]
Hence, this simplified equation represents the tangent line at the given point.

When dealing with inverses, understanding the connection between a function and its inverse is crucial. An important property is how slopes are related at corresponding points. For the original function \( f \) and its inverse \( f^{-1} \), the slopes are reciprocals if the function is differentiable and its inverse is differentiable at those points.
Given that the slope of \( f(s) \) at \( s = 9 \) is \( \frac{1}{6} \), the corresponding point on the inverse function \( f^{-1} \) has coordinates \((f(9), 9) = (3, 9)\). The slope of the tangent to the inverse, \( f^{-1}(y) \), at \( y = 3 \) is found by taking the reciprocal of the slope of \( f \) at \( s = 9 \), resulting in \( 6 \).
This reciprocal relationship arises due to the symmetry of inverse functions in the context of their graphical representation, typically reflecting over the line \( y = x \). This concept is vital under the assumptions that \( f \) is strictly monotonic, ensuring the existence of a continuous inverse function.