How do you use the definition of a derivative to find the derivative of #f(x)=4+x-2x^2#?

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Answer 1 · 2016-05-12T08:00:06

#(df)/(dx)==1-4x#

Derivative of function #f(x)# is defined as

#(df)/(dx)=Lt_(Deltax->0)(f(x+Deltax)-f(x))/(Deltax)#

As #f(x)=4+x-2x^2#

#f(x+Deltax)=4+x+Deltax-2(x+Deltax)^2# or

#f(x+Deltax)=4+x+Deltax-2(x^2+2xDeltax+(Deltax)^2)# or

#f(x+Deltax)=4+x+Deltax-2x^2-4xDeltax+2(Deltax)^2#

Hence, #f(x+Deltax)-f(x)=Deltax-4xDeltax+2(Deltax)^2#

i.e. #(f(x+Deltax)-f(x))/(Deltax)=1-4x+2Deltax#

Hence, #(df)/(dx)=Lt_(Deltax->0)(1-4x+2Deltax)=1-4x#