007A Sample Final 3, Problem 3 Detailed Solution

Find the derivative of the following function using the limit definition of the derivative:

$f(x)=3x-x^{2}$

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Background Information:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}}

Solution:

Step 1:
Using the limit definition of derivative, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{f'(x)} & = & \displaystyle{\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} \frac{(3(x+h)-(x+h)^2)-(3x-x^2)}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3x+3h-(x^2+2xh+h^2)-3x+x^2}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3h-2xh-h^2}{h}.} \end{array}}

Step 2:
Now, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{f'(x)} & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3h-2xh-h^2}{h}}\\ &&\\ & = & \displaystyle{\lim_{h\rightarrow 0} 3-2x-h}\\ &&\\ & = & \displaystyle{3-2x.} \end{array}}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=3-2x}

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