Question

By the first principle (The definition of Derivative) to find the derivative of f(x)= x−2 2x+3 ​ .

Answer 1

The derivative of
`\( f(x) = (x - 2)/(2x + 3) \)` using the first principle (definition of derivative) is
`\( f'(x) = (-4x - 13)/((2x + 3)^2) \)`.

Step-by-step explanation:

Derivatives express the rate of change of a function at a specific point. The first principle, defining a derivative, involves using the limit definition:
`\( f'(x) = \lim_{{h \to 0}} (f(x+h) - f(x))/(h) \)`. Applying this to
`\( f(x) = (x - 2)/(2x + 3) \)`, we substitute
`\( f(x) \)` into the formula.

First,
`\( f(x + h) \) and \( f(x) \)` are computed:
`\( f(x + h) = (x + h - 2)/(2(x + h) + 3) \) and \( f(x) = (x - 2)/(2x + 3) \)`.

Then,
`\( f(x + h) - f(x) \)` is found and simplified, resulting in
`\( ((x + h - 2)(2x + 3) - (x - 2)(2(x + h) + 3))/((2(x + h) + 3)(2x + 3)) \)`.

After simplifying the numerator, the limit definition is applied:
`\( f'(x) = \lim_{{h \to 0}} (-4x - 13)/((2x + 3)^2) \)`.

This derivative,
`\( f'(x) = (-4x - 13)/((2x + 3)^2) \)`, represents the rate of change of
`\( f(x) \)` at any point
`\( x \)`, showcasing the function's slope at that specific
`\( x \)` value.