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Use the alternative form of the derivative to find the derivative at x=c (if it exists).f(x) = 3/x, c=4


Answered 9 months ago
Answered 9 months ago
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To solve this question, we use the alternative form of the derivative.

f(x)=limxcf(x)f(c)xc\begin{align*} f'(x)=\lim_{x\rightarrow c}{\dfrac{f(x)-f(c)}{x-c}} \end{align*}

Using this we have

f(4)=limx4f(x)f(4)x4=limx43x34x41\begin{align*} f'(4)&=\lim_{x\rightarrow 4}{\dfrac{f(x)-f(4)}{x-4}}\\\\ &=\lim_{x\rightarrow 4}{\dfrac{\dfrac{3}{x}-\dfrac{3}{4}}{\dfrac{x-4}{1}}}\tag{1} \end{align*}

Because fractions in numerator have different denominators the first step is to find equivalent fractions which have common denominator. We find the Least Common Denominator (LCD) by multiplying denominators: 4x4x then rewrite fractions in the equation as equivalent fractions using the LCD as denominator. Then add and simplify.

(1)=limx4433x4xx41=limx43(x4)4xx41=limx43(x4)4x(x4)=limx434x=344=316\begin{align*} (1)&=\lim_{x\rightarrow 4}{\dfrac{\dfrac{4\cdot3-3x}{4x}}{\dfrac{x-4}{1}}} \\\\ &=\lim_{x\rightarrow 4}{\dfrac{\dfrac{-3(x-4)}{4x}}{\dfrac{x-4}{1}}}\tag{Factor out -3.}\\\\ &=\lim_{x\rightarrow 4}{\dfrac{-3\cancel{(x-4)}}{4x\cancel{(x-4)}}}\tag{Simplify complex fraction.}\\\\ &=\lim_{x\rightarrow 4}{\dfrac{-3}{4x}}\\\\ &=\dfrac{-3}{4\cdot 4}\\\\ &={\color{Mahogany}\dfrac{-3}{16}} \end{align*}

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