Question

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{f(x)=\frac{x+1}{2 x-3}} \quad \frac{\text { Point }}{(2,3)}\)

Short answer

The derivative of the function \(f(x) = \frac{x + 1}{2x - 3}\) evaluated at the point (2, 3) is -5.

Step-by-step solution

Identify The Component Functions

In the given function \(f(x) = \frac{x + 1}{2x - 3}\), identify the component functions for the Quotient Rule. Here, \(u(x) = x + 1\) and \(v(x) = 2x - 3\).

Calculate The Derivatives Of Component Functions

Calculate the derivatives of \(u(x)\) and \(v(x)\). Using the power rule for differentiation, you will find that \(u'(x) = 1\) and \(v'(x) = 2\).

Apply The Quotient Rule

Having the required functions and their derivatives, apply the Quotient Rule. The derivative of the function \(f\), \(f'(x)\), is given by: \(f'(x) = \frac{(2x-3)(1)-(x+1)(2)}{(2x-3)^2} = \frac{2x - 3 - 2x - 2}{(2x - 3)^2} = \frac{-5}{(2x - 3)^2}\).

Evaluate The Derivative At The Given Point

Substitute \(x = 2\) into \(f'(x) = \frac{-5}{(2x - 3)^2}\) to find the derivative at the given point. Therefore, \(f'(2) = \frac{-5}{(2*2 - 3)^2} = -5\).