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Chapter 3: Problem 66

Use any method to evaluate the derivative of the following functions. $$f(x)=\frac{4-x^{2}}{x-2}$$

Short Answer

Expert verified
Question: Determine the derivative of the function $$f(x)=\frac{4-x^{2}}{x-2}$$ using the quotient rule. Answer: The derivative of the function is $$f'(x)=\frac{-x^2+4x-4}{(x-2)^2}$$.

Step by step solution

01

Identify u and v

First, we identify the functions u and v in the given function. We have: $$u(x) =4-x^2$$ $$v(x) =x-2$$
02

Compute u' and v'

Now, we compute the derivatives of u and v with respect to x: $$u'(x) = -2x$$ $$v'(x) = 1$$
03

Apply the Quotient Rule

Using the quotient rule for derivatives, we get: $$f'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{(v(x))^2}$$ Substituting our values for u, v, u', and v': $$f'(x)=\frac{(-2x)(x-2)-(4-x^2)(1)}{(x-2)^2}$$
04

Simplify the expression

Now, we simplify the expression for the derivative: $$f'(x)=\frac{-2x^2+4x-4+x^2}{(x-2)^2}$$ $$f'(x)=\frac{-x^2+4x-4}{(x-2)^2}$$
05

Final answer

So, the derivative of the given function is: $$f'(x)=\frac{-x^2+4x-4}{(x-2)^2}$$

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