Finding the Derivative of a Function at a Number \(c\)

Find the derivative of the function \(f( x) =x^{2}-5x\) at any real number \(c\) using form (2).

Solution Using form (2), we have \begin{eqnarray*} f^\prime ( c) &=&\lim\limits_{h\rightarrow 0}\dfrac{f(c+h) -f( c) }{h} = \lim\limits_{h\rightarrow 0}\dfrac{[ ( c+h) ^{2}-5( c+h) ] -( c^{2}-5c) }{h} \\[4pt] \notag &=&\lim\limits_{h\rightarrow 0}\dfrac{[ ( c^{2}+2ch+h^{2}) -5c-5h] -c^{2}+5c}{h}=\lim\limits_{h\rightarrow 0}\dfrac{2ch+h^{2}-5h}{ h} \\[4pt] \notag &=&\lim\limits_{h\rightarrow 0}\dfrac{h( 2c+h-5) }{h} =\lim\limits_{h\rightarrow 0}( 2c+h-5) =2c-5 \end{eqnarray*}