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:
$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}$

Solution:

Step 1:
Using the limit definition of derivative, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {(3(x+h)-(x+h)^{2})-(3x-x^{2})}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3x+3h-(x^{2}+2xh+h^{2})-3x+x^{2}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3h-2xh-h^{2}}{h}}.}\end{array}}$

Step 2:
Now, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\lim _{h\rightarrow 0}{\frac {3h-2xh-h^{2}}{h}}}\\&&\\&=&\displaystyle {\lim _{h\rightarrow 0}3-2x-h}\\&&\\&=&\displaystyle {3-2x.}\end{array}}$

Final Answer:
$f'(x)=3-2x$

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