MathAdmin: Created page with "Find the derivative of the following function using the limit definition of the derivative: :: $f(x)=3x-x^2$ {| class="mw-col..."

2017-04-10T16:29:07Z

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

::<math>f(x)=3x-x^2</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|<math style="vertical-align: -13px">f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}</math>
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|Using the limit definition of derivative, we have
|-
|        <math>\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}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Now, we have
|-
|        <math>\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}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|       <math>f'(x)=3-2x</math>
|}
[[009A_Sample_Final_3|'''Return to Sample Exam''']]

009A Sample Final 3, Problem 3 - Revision history

MathAdmin: Created page with "Find the derivative of the following function using the limit definition of the derivative: ::f(x)=3x-x^2 {| class="mw-col..."

MathAdmin: Created page with "Find the derivative of the following function using the limit definition of the derivative: :: $f(x)=3x-x^2$ {| class="mw-col..."