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

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