Finding the Derivative of a Function at a Number

Find the derivative of \(f(x) =2x^{3}-5x\) at \(x=1\). That is, find \(f^\prime (1) \).

Solution Using the definition of a derivative, we have \[ \begin{array}{@{\hspace*{-4pc}}rcl} f^\prime (1) &=&\lim\limits_{x\rightarrow 1}\dfrac{f( x) -f(1) }{x-1}=\lim\limits_{x\rightarrow 1}\dfrac{( 2x^{3}-5x) -( -3) }{x-1}=\lim\limits_{x\rightarrow 1}\dfrac{ 2x^{3}-5x+3}{x-1} \quad {\color{#0066A7}{\hbox{\(f(1) =2-5=-3\)}}} \\ &=&\lim\limits_{x\rightarrow 1}\dfrac{( x-1) ( 2x^{2}+2x-3) }{x-1}=\lim\limits_{x\rightarrow 1}(2x^{2}+2x-3) =2+2-3=1 \end{array} \]