Questions: 1. f(x)=(x^2+2x-4)/(2x) find derivative

Transcript text: 1. $f(x)=\frac{x^{2}+2 x-4}{2 x}$ find derivative

Solution

Solution Steps

Step 1: Find the Derivatives

We start by identifying the functions $ g(x) $ and $ h(x) $ from the given function $ f(x) = \frac{g(x)}{h(x)} $, where:

$ g(x) = x^2 + 2x - 4 $
$ h(x) = 2x $

Next, we compute their derivatives:

$ g'(x) = 2x + 2 $
$ h'(x) = 2 $

Step 2: Apply the Quotient Rule

Using the quotient rule, we find the derivative $ f'(x) $: \[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \] Substituting the derivatives and functions: \[ f'(x) = \frac{(2x + 2)(2x) - (x^2 + 2x - 4)(2)}{(2x)^2} \]

Step 3: Simplify the Expression

After substituting and simplifying the expression, we arrive at: \[ f'(x) = \frac{-2x^2 + 2x(2x + 2) - 4x + 8}{4x^2} \] This simplifies further to: \[ f'(x) = \frac{1}{2} + \frac{2}{x^2} \]

Final Answer

The derivative of the function $ f(x) $ is: \[ \boxed{f'(x) = \frac{1}{2} + \frac{2}{x^2}} \]

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